L(s) = 1 | + 2-s + 9·4-s − 24·7-s + 25·8-s + 22·11-s − 30·13-s − 24·14-s + 41·16-s + 106·17-s + 50·19-s + 22·22-s + 134·23-s − 30·26-s − 216·28-s + 198·29-s + 360·31-s + 249·32-s + 106·34-s + 328·37-s + 50·38-s + 782·41-s − 386·43-s + 198·44-s + 134·46-s + 266·47-s + 134·49-s − 270·52-s + ⋯ |
L(s) = 1 | + 0.353·2-s + 9/8·4-s − 1.29·7-s + 1.10·8-s + 0.603·11-s − 0.640·13-s − 0.458·14-s + 0.640·16-s + 1.51·17-s + 0.603·19-s + 0.213·22-s + 1.21·23-s − 0.226·26-s − 1.45·28-s + 1.26·29-s + 2.08·31-s + 1.37·32-s + 0.534·34-s + 1.45·37-s + 0.213·38-s + 2.97·41-s − 1.36·43-s + 0.678·44-s + 0.429·46-s + 0.825·47-s + 0.390·49-s − 0.720·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.111274150\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.111274150\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - T - p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 30 T + 4522 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 106 T + 7882 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 50 T + 14246 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 134 T + 26398 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 328 T + 62630 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 782 T + 285970 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 386 T + 179870 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 266 T + 92542 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 522 T + 295162 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 776 T + 528582 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1296 T + 1178926 T^{2} + 1296 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 652 T + 589506 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 324 T + 579670 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 452 T + 982470 T^{2} - 452 p^{3} T^{3} + p^{6} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.795759133585842790037926363210, −8.261579963786393068971411877193, −7.80968275803579131701313037068, −7.51235855780690021742990014837, −7.31943777769190295922530629347, −6.73778581700204188735748099078, −6.45017574975322343354199701980, −6.11976253167072302098022074699, −5.90904704424198216711065960583, −5.24107771260028231703855754604, −4.62111246498442470999945091099, −4.59061766979703980579529316556, −3.95696909420329897397791955117, −3.27799926686015587703525492087, −2.93943365083729515510651542215, −2.77023486848679734207366342686, −2.25404598355844444123514292016, −1.27496928258631777666068177414, −1.11018526135846204203044094696, −0.57126054679561116444234669376,
0.57126054679561116444234669376, 1.11018526135846204203044094696, 1.27496928258631777666068177414, 2.25404598355844444123514292016, 2.77023486848679734207366342686, 2.93943365083729515510651542215, 3.27799926686015587703525492087, 3.95696909420329897397791955117, 4.59061766979703980579529316556, 4.62111246498442470999945091099, 5.24107771260028231703855754604, 5.90904704424198216711065960583, 6.11976253167072302098022074699, 6.45017574975322343354199701980, 6.73778581700204188735748099078, 7.31943777769190295922530629347, 7.51235855780690021742990014837, 7.80968275803579131701313037068, 8.261579963786393068971411877193, 8.795759133585842790037926363210