| L(s) = 1 | + 7·3-s + 25·5-s + 20·7-s − 28·9-s + 63·11-s − 99·13-s + 175·15-s − 44·17-s + 199·19-s + 140·21-s − 115·23-s + 375·25-s − 364·27-s + 231·29-s + 518·31-s + 441·33-s + 500·35-s − 113·37-s − 693·39-s − 174·41-s + 298·43-s − 700·45-s + 360·47-s − 739·49-s − 308·51-s + 217·53-s + 1.57e3·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 2.23·5-s + 1.07·7-s − 1.03·9-s + 1.72·11-s − 2.11·13-s + 3.01·15-s − 0.627·17-s + 2.40·19-s + 1.45·21-s − 1.04·23-s + 3·25-s − 2.59·27-s + 1.47·29-s + 3.00·31-s + 2.32·33-s + 2.41·35-s − 0.502·37-s − 2.84·39-s − 0.662·41-s + 1.05·43-s − 2.31·45-s + 1.11·47-s − 2.15·49-s − 0.845·51-s + 0.562·53-s + 3.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(70.15714030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(70.15714030\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{5} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 - 7 T + 77 T^{2} - 371 T^{3} + 905 p T^{4} - 3280 p T^{5} + 905 p^{4} T^{6} - 371 p^{6} T^{7} + 77 p^{9} T^{8} - 7 p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 20 T + 1139 T^{2} - 15003 T^{3} + 82368 p T^{4} - 5857022 T^{5} + 82368 p^{4} T^{6} - 15003 p^{6} T^{7} + 1139 p^{9} T^{8} - 20 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 63 T + 4852 T^{2} - 17352 p T^{3} + 10911079 T^{4} - 357603786 T^{5} + 10911079 p^{3} T^{6} - 17352 p^{7} T^{7} + 4852 p^{9} T^{8} - 63 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 99 T + 9409 T^{2} + 609459 T^{3} + 34133101 T^{4} + 1707861774 T^{5} + 34133101 p^{3} T^{6} + 609459 p^{6} T^{7} + 9409 p^{9} T^{8} + 99 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 44 T + 777 p T^{2} + 37327 p T^{3} + 98129218 T^{4} + 4562214742 T^{5} + 98129218 p^{3} T^{6} + 37327 p^{7} T^{7} + 777 p^{10} T^{8} + 44 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 199 T + 29822 T^{2} - 3037304 T^{3} + 275935505 T^{4} - 22140975770 T^{5} + 275935505 p^{3} T^{6} - 3037304 p^{6} T^{7} + 29822 p^{9} T^{8} - 199 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 231 T + 78598 T^{2} - 5649961 T^{3} + 1205054479 T^{4} + 42470639048 T^{5} + 1205054479 p^{3} T^{6} - 5649961 p^{6} T^{7} + 78598 p^{9} T^{8} - 231 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 518 T + 240348 T^{2} - 67378123 T^{3} + 17017081385 T^{4} - 3086057641785 T^{5} + 17017081385 p^{3} T^{6} - 67378123 p^{6} T^{7} + 240348 p^{9} T^{8} - 518 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 113 T + 53199 T^{2} + 10949488 T^{3} + 4012092620 T^{4} + 120172958958 T^{5} + 4012092620 p^{3} T^{6} + 10949488 p^{6} T^{7} + 53199 p^{9} T^{8} + 113 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 174 T + 279232 T^{2} + 46368997 T^{3} + 34537283767 T^{4} + 4751319154867 T^{5} + 34537283767 p^{3} T^{6} + 46368997 p^{6} T^{7} + 279232 p^{9} T^{8} + 174 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 298 T + 190899 T^{2} - 59133608 T^{3} + 24377195150 T^{4} - 6439036031100 T^{5} + 24377195150 p^{3} T^{6} - 59133608 p^{6} T^{7} + 190899 p^{9} T^{8} - 298 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 360 T + 58312 T^{2} - 10023010 T^{3} + 16238168623 T^{4} - 7876318719388 T^{5} + 16238168623 p^{3} T^{6} - 10023010 p^{6} T^{7} + 58312 p^{9} T^{8} - 360 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 217 T + 319213 T^{2} - 61542500 T^{3} + 63757793998 T^{4} - 12781053358422 T^{5} + 63757793998 p^{3} T^{6} - 61542500 p^{6} T^{7} + 319213 p^{9} T^{8} - 217 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 1551 T + 1803809 T^{2} - 1407030072 T^{3} + 901101397628 T^{4} - 446206985406594 T^{5} + 901101397628 p^{3} T^{6} - 1407030072 p^{6} T^{7} + 1803809 p^{9} T^{8} - 1551 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 737 T + 991288 T^{2} + 614487378 T^{3} + 418410409351 T^{4} + 203323723845650 T^{5} + 418410409351 p^{3} T^{6} + 614487378 p^{6} T^{7} + 991288 p^{9} T^{8} + 737 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 539 T + 312261 T^{2} + 37406200 T^{3} + 68950977400 T^{4} - 17699993425578 T^{5} + 68950977400 p^{3} T^{6} + 37406200 p^{6} T^{7} + 312261 p^{9} T^{8} - 539 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 1736 T + 2099380 T^{2} - 1522133061 T^{3} + 988591941185 T^{4} - 536636656278389 T^{5} + 988591941185 p^{3} T^{6} - 1522133061 p^{6} T^{7} + 2099380 p^{9} T^{8} - 1736 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 628 T + 1090244 T^{2} + 745063200 T^{3} + 640976563959 T^{4} + 403412613706192 T^{5} + 640976563959 p^{3} T^{6} + 745063200 p^{6} T^{7} + 1090244 p^{9} T^{8} + 628 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 2954 T + 5473759 T^{2} - 7030130328 T^{3} + 7021210589230 T^{4} - 5487141989090876 T^{5} + 7021210589230 p^{3} T^{6} - 7030130328 p^{6} T^{7} + 5473759 p^{9} T^{8} - 2954 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 153 T + 705247 T^{2} - 466954664 T^{3} - 14297173262 T^{4} - 404840525700878 T^{5} - 14297173262 p^{3} T^{6} - 466954664 p^{6} T^{7} + 705247 p^{9} T^{8} - 153 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 1558 T + 1516081 T^{2} - 114748648 T^{3} - 1031601851306 T^{4} - 1529677351065660 T^{5} - 1031601851306 p^{3} T^{6} - 114748648 p^{6} T^{7} + 1516081 p^{9} T^{8} + 1558 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 1375 T + 2424902 T^{2} + 2520846068 T^{3} + 3336288405557 T^{4} + 2658583838352338 T^{5} + 3336288405557 p^{3} T^{6} + 2520846068 p^{6} T^{7} + 2424902 p^{9} T^{8} + 1375 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.05267723617911910616782628432, −4.91456947452859141053200601824, −4.87961342613089504432441792998, −4.80235164600138140334194591249, −4.48307354007271912063392064259, −4.43647815028406431017084884389, −3.87452172003777083667304973368, −3.82271866289630784398822653776, −3.79662060782058097063943939827, −3.53148459000929883645390454409, −3.04609205388488149154337295478, −2.88929986535424942238605655164, −2.84117441667152086562684941080, −2.76505820303243028273320696130, −2.72549893785988672425757666563, −2.08109274646298116680751870412, −2.07930434766038296564802902816, −2.02791405059393351562158624898, −1.88448872704778102829781427168, −1.56579026768293025970836589827, −1.10569076546851158188970254640, −1.01669220867529933462426184054, −0.70502535921906902641467237150, −0.57332705008364078458165583906, −0.43843374369526292613016546569,
0.43843374369526292613016546569, 0.57332705008364078458165583906, 0.70502535921906902641467237150, 1.01669220867529933462426184054, 1.10569076546851158188970254640, 1.56579026768293025970836589827, 1.88448872704778102829781427168, 2.02791405059393351562158624898, 2.07930434766038296564802902816, 2.08109274646298116680751870412, 2.72549893785988672425757666563, 2.76505820303243028273320696130, 2.84117441667152086562684941080, 2.88929986535424942238605655164, 3.04609205388488149154337295478, 3.53148459000929883645390454409, 3.79662060782058097063943939827, 3.82271866289630784398822653776, 3.87452172003777083667304973368, 4.43647815028406431017084884389, 4.48307354007271912063392064259, 4.80235164600138140334194591249, 4.87961342613089504432441792998, 4.91456947452859141053200601824, 5.05267723617911910616782628432
