| L(s) = 1 | + 4·3-s + 2·4-s + 6·9-s + 6·11-s + 8·12-s + 12·13-s + 16-s − 16·17-s − 6·19-s − 4·23-s − 8·29-s + 24·33-s + 12·36-s − 30·37-s + 48·39-s − 14·43-s + 12·44-s + 4·48-s − 7·49-s − 64·51-s + 24·52-s − 16·53-s − 24·57-s + 24·59-s − 16·61-s − 2·64-s − 24·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4-s + 2·9-s + 1.80·11-s + 2.30·12-s + 3.32·13-s + 1/4·16-s − 3.88·17-s − 1.37·19-s − 0.834·23-s − 1.48·29-s + 4.17·33-s + 2·36-s − 4.93·37-s + 7.68·39-s − 2.13·43-s + 1.80·44-s + 0.577·48-s − 49-s − 8.96·51-s + 3.32·52-s − 2.19·53-s − 3.17·57-s + 3.12·59-s − 2.04·61-s − 1/4·64-s − 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0003199537149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0003199537149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 12 T + 45 T^{2} + 24 T^{3} - 556 T^{4} + 24 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 + p T^{2} - 3 T^{4} - 130 T^{6} + 1248 T^{7} - 754 T^{8} + 1248 p T^{9} - 130 p^{2} T^{10} - 3 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 6 T + 26 T^{2} - 84 T^{3} + 93 T^{4} - 216 T^{5} + 214 T^{6} + 366 T^{7} + 9716 T^{8} + 366 p T^{9} + 214 p^{2} T^{10} - 216 p^{3} T^{11} + 93 p^{4} T^{12} - 84 p^{5} T^{13} + 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + 217 T^{4} - 2234 T^{6} - 6492 T^{7} - 40022 T^{8} - 6492 p T^{9} - 2234 p^{2} T^{10} + 217 p^{4} T^{12} + 6 p^{6} T^{13} + 31 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 4 T - 33 T^{2} - 324 T^{3} + 113 T^{4} + 8736 T^{5} + 34370 T^{6} - 109208 T^{7} - 1157922 T^{8} - 109208 p T^{9} + 34370 p^{2} T^{10} + 8736 p^{3} T^{11} + 113 p^{4} T^{12} - 324 p^{5} T^{13} - 33 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 8 T - 13 T^{2} - 480 T^{3} - 1615 T^{4} + 7528 T^{5} + 59362 T^{6} - 2520 T^{7} - 1294994 T^{8} - 2520 p T^{9} + 59362 p^{2} T^{10} + 7528 p^{3} T^{11} - 1615 p^{4} T^{12} - 480 p^{5} T^{13} - 13 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 74 T^{2} + 3321 T^{4} - 90346 T^{6} + 2694452 T^{8} - 90346 p^{2} T^{10} + 3321 p^{4} T^{12} - 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 + 30 T + 514 T^{2} + 6420 T^{3} + 64101 T^{4} + 541620 T^{5} + 4034198 T^{6} + 27314130 T^{7} + 171690740 T^{8} + 27314130 p T^{9} + 4034198 p^{2} T^{10} + 541620 p^{3} T^{11} + 64101 p^{4} T^{12} + 6420 p^{5} T^{13} + 514 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 + 80 T^{2} + 2370 T^{4} - 7488 T^{5} + 65728 T^{6} - 773760 T^{7} + 3160547 T^{8} - 773760 p T^{9} + 65728 p^{2} T^{10} - 7488 p^{3} T^{11} + 2370 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 + 14 T + 67 T^{2} - 134 T^{3} - 3763 T^{4} - 24232 T^{5} - 46294 T^{6} + 310492 T^{7} + 3087010 T^{8} + 310492 p T^{9} - 46294 p^{2} T^{10} - 24232 p^{3} T^{11} - 3763 p^{4} T^{12} - 134 p^{5} T^{13} + 67 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 148 T^{2} + 14490 T^{4} - 1021616 T^{6} + 54203939 T^{8} - 1021616 p^{2} T^{10} + 14490 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 8 T + 137 T^{2} + 1268 T^{3} + 8956 T^{4} + 1268 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 24 T + 475 T^{2} - 6792 T^{3} + 87253 T^{4} - 932736 T^{5} + 9243862 T^{6} - 80308224 T^{7} + 654943486 T^{8} - 80308224 p T^{9} + 9243862 p^{2} T^{10} - 932736 p^{3} T^{11} + 87253 p^{4} T^{12} - 6792 p^{5} T^{13} + 475 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 8 T - 62 T^{2} + 32 T^{3} + 8251 T^{4} + 32 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 24 T + 432 T^{2} + 5760 T^{3} + 69970 T^{4} + 776424 T^{5} + 7801920 T^{6} + 72545640 T^{7} + 609479859 T^{8} + 72545640 p T^{9} + 7801920 p^{2} T^{10} + 776424 p^{3} T^{11} + 69970 p^{4} T^{12} + 5760 p^{5} T^{13} + 432 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 12 T + 128 T^{2} + 960 T^{3} + 5202 T^{4} - 11124 T^{5} - 245984 T^{6} - 4504236 T^{7} - 41845549 T^{8} - 4504236 p T^{9} - 245984 p^{2} T^{10} - 11124 p^{3} T^{11} + 5202 p^{4} T^{12} + 960 p^{5} T^{13} + 128 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 192 T^{2} + 25276 T^{4} - 2537280 T^{6} + 214285254 T^{8} - 2537280 p^{2} T^{10} + 25276 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 10 T + 169 T^{2} + 1510 T^{3} + 17728 T^{4} + 1510 p T^{5} + 169 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 208 T^{2} + 27708 T^{4} - 2708144 T^{6} + 235978406 T^{8} - 2708144 p^{2} T^{10} + 27708 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 42 T + 1115 T^{2} - 22134 T^{3} + 362637 T^{4} - 5064156 T^{5} + 62362882 T^{6} - 684879408 T^{7} + 6796999778 T^{8} - 684879408 p T^{9} + 62362882 p^{2} T^{10} - 5064156 p^{3} T^{11} + 362637 p^{4} T^{12} - 22134 p^{5} T^{13} + 1115 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 24 T + 408 T^{2} - 5184 T^{3} + 49906 T^{4} - 311304 T^{5} + 4896 T^{6} + 24434760 T^{7} - 344198589 T^{8} + 24434760 p T^{9} + 4896 p^{2} T^{10} - 311304 p^{3} T^{11} + 49906 p^{4} T^{12} - 5184 p^{5} T^{13} + 408 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.69117230279806945662357127693, −3.66307500441253365764896145994, −3.62993990458785228971617107922, −3.58612472910013080188858331927, −3.54196643674786248372702769150, −3.51943283146633802286449440097, −3.10946322447399766636850382562, −2.85560236302605409194866738618, −2.85541899615371069870083059123, −2.78270509646176347915409655739, −2.52377102198216308430961333583, −2.33892474589656356350513027110, −2.30236746574281210099823021568, −2.20336296424896633306526608972, −2.11540403919705122709696245699, −1.91621691774840402290996696314, −1.66124831652593353105429360434, −1.61795694135270102328332582110, −1.53304504549704020703507797225, −1.42808195887286054505469173630, −1.28186138956945485334576995110, −1.23335174190037613221964578843, −0.67087426313688332509612263100, −0.10229906654547800742281297111, −0.00362038470875086783940575798,
0.00362038470875086783940575798, 0.10229906654547800742281297111, 0.67087426313688332509612263100, 1.23335174190037613221964578843, 1.28186138956945485334576995110, 1.42808195887286054505469173630, 1.53304504549704020703507797225, 1.61795694135270102328332582110, 1.66124831652593353105429360434, 1.91621691774840402290996696314, 2.11540403919705122709696245699, 2.20336296424896633306526608972, 2.30236746574281210099823021568, 2.33892474589656356350513027110, 2.52377102198216308430961333583, 2.78270509646176347915409655739, 2.85541899615371069870083059123, 2.85560236302605409194866738618, 3.10946322447399766636850382562, 3.51943283146633802286449440097, 3.54196643674786248372702769150, 3.58612472910013080188858331927, 3.62993990458785228971617107922, 3.66307500441253365764896145994, 3.69117230279806945662357127693
Plot not available for L-functions of degree greater than 10.
