| L(s) = 1 | + 12·3-s − 8·4-s + 28·7-s + 3·8-s + 90·9-s + 21·11-s − 96·12-s + 5·13-s + 18·16-s + 99·17-s + 72·19-s + 336·21-s + 102·23-s + 36·24-s + 540·27-s − 224·28-s − 240·29-s + 351·31-s − 24·32-s + 252·33-s − 720·36-s + 399·37-s + 60·39-s + 381·41-s + 460·43-s − 168·44-s + 60·47-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4-s + 1.51·7-s + 0.132·8-s + 10/3·9-s + 0.575·11-s − 2.30·12-s + 0.106·13-s + 9/32·16-s + 1.41·17-s + 0.869·19-s + 3.49·21-s + 0.924·23-s + 0.306·24-s + 3.84·27-s − 1.51·28-s − 1.53·29-s + 2.03·31-s − 0.132·32-s + 1.32·33-s − 3.33·36-s + 1.77·37-s + 0.246·39-s + 1.45·41-s + 1.63·43-s − 0.575·44-s + 0.186·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(32.28738989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.28738989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{3} T^{2} - 3 T^{3} + 23 p T^{4} - 3 p^{3} T^{5} + p^{9} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 21 T + 2681 T^{2} - 22236 T^{3} + 3289690 T^{4} - 22236 p^{3} T^{5} + 2681 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 1848 T^{2} + 69653 T^{3} - 988138 T^{4} + 69653 p^{3} T^{5} + 1848 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 99 T + 13298 T^{2} - 508917 T^{3} + 56706418 T^{4} - 508917 p^{3} T^{5} + 13298 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 72 T + 25164 T^{2} - 1424592 T^{3} + 252125750 T^{4} - 1424592 p^{3} T^{5} + 25164 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 102 T + 38528 T^{2} - 3477336 T^{3} + 641077657 T^{4} - 3477336 p^{3} T^{5} + 38528 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 240 T + 86430 T^{2} + 14441664 T^{3} + 3156579983 T^{4} + 14441664 p^{3} T^{5} + 86430 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 351 T + 102010 T^{2} - 21726711 T^{3} + 4440563682 T^{4} - 21726711 p^{3} T^{5} + 102010 p^{6} T^{6} - 351 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 399 T + 123385 T^{2} - 25924734 T^{3} + 6891314910 T^{4} - 25924734 p^{3} T^{5} + 123385 p^{6} T^{6} - 399 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 381 T + 273692 T^{2} - 71526783 T^{3} + 28134293782 T^{4} - 71526783 p^{3} T^{5} + 273692 p^{6} T^{6} - 381 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 460 T + 241338 T^{2} - 78643232 T^{3} + 29430041627 T^{4} - 78643232 p^{3} T^{5} + 241338 p^{6} T^{6} - 460 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 60 T + 130548 T^{2} - 11848116 T^{3} + 16872903254 T^{4} - 11848116 p^{3} T^{5} + 130548 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 873 T + 701906 T^{2} - 368908875 T^{3} + 161980949146 T^{4} - 368908875 p^{3} T^{5} + 701906 p^{6} T^{6} - 873 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 855 T + 767616 T^{2} + 344685519 T^{3} + 195821792246 T^{4} + 344685519 p^{3} T^{5} + 767616 p^{6} T^{6} + 855 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 687 T + 715504 T^{2} - 385624593 T^{3} + 235308406830 T^{4} - 385624593 p^{3} T^{5} + 715504 p^{6} T^{6} - 687 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 503 T + 6753 p T^{2} + 193160692 T^{3} + 211333484900 T^{4} + 193160692 p^{3} T^{5} + 6753 p^{7} T^{6} + 503 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 681 T + 1310019 T^{2} + 668258544 T^{3} + 678493576820 T^{4} + 668258544 p^{3} T^{5} + 1310019 p^{6} T^{6} + 681 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1228 T + 618972 T^{2} + 228294532 T^{3} - 319091414218 T^{4} + 228294532 p^{3} T^{5} + 618972 p^{6} T^{6} - 1228 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 345 T + 1235121 T^{2} - 721148448 T^{3} + 716751093026 T^{4} - 721148448 p^{3} T^{5} + 1235121 p^{6} T^{6} - 345 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1509 T + 2187990 T^{2} - 1588008513 T^{3} + 1474125264218 T^{4} - 1588008513 p^{3} T^{5} + 2187990 p^{6} T^{6} - 1509 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 198 T + 1184884 T^{2} - 492558966 T^{3} + 650980125750 T^{4} - 492558966 p^{3} T^{5} + 1184884 p^{6} T^{6} + 198 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 372 T + 1910724 T^{2} + 1429297356 T^{3} + 2012229372278 T^{4} + 1429297356 p^{3} T^{5} + 1910724 p^{6} T^{6} + 372 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50179620900643305834993893748, −7.42435159833599397914670702253, −7.35095989976259354814582132288, −6.68335271674406926938209006426, −6.60909373085913891498968973311, −6.12431002378088758982728224238, −5.79103803939011231087157379419, −5.73390103094970890650081308637, −5.42816531248477488642199228366, −4.81634607216087304675965402979, −4.81339307417873964966671028383, −4.58354015178327411564346441714, −4.38919649214022506706923322551, −3.95760365285180002592721842104, −3.76438616926355740433122976364, −3.59606247675145698222919911121, −3.14319674661012380147651949627, −2.91798430453661947416398156307, −2.57923575051564932741480863025, −2.17938860412614093740260228562, −2.04656329438411289135604516172, −1.51595956588549286391492374356, −0.980134954354101282695724802418, −0.851715233936014343985784508710, −0.78400160954035339870385112698,
0.78400160954035339870385112698, 0.851715233936014343985784508710, 0.980134954354101282695724802418, 1.51595956588549286391492374356, 2.04656329438411289135604516172, 2.17938860412614093740260228562, 2.57923575051564932741480863025, 2.91798430453661947416398156307, 3.14319674661012380147651949627, 3.59606247675145698222919911121, 3.76438616926355740433122976364, 3.95760365285180002592721842104, 4.38919649214022506706923322551, 4.58354015178327411564346441714, 4.81339307417873964966671028383, 4.81634607216087304675965402979, 5.42816531248477488642199228366, 5.73390103094970890650081308637, 5.79103803939011231087157379419, 6.12431002378088758982728224238, 6.60909373085913891498968973311, 6.68335271674406926938209006426, 7.35095989976259354814582132288, 7.42435159833599397914670702253, 7.50179620900643305834993893748
