| L(s) = 1 | − 54·3-s − 52·5-s + 686·7-s + 2.18e3·9-s − 4.05e3·11-s − 6.46e3·13-s + 2.80e3·15-s + 7.78e3·17-s + 1.54e4·19-s − 3.70e4·21-s + 1.03e5·23-s − 3.26e4·25-s − 7.87e4·27-s + 1.02e5·29-s − 7.17e4·31-s + 2.19e5·33-s − 3.56e4·35-s − 3.82e5·37-s + 3.49e5·39-s − 6.23e5·41-s − 6.47e5·43-s − 1.13e5·45-s + 8.02e5·47-s + 3.52e5·49-s − 4.20e5·51-s − 2.02e6·53-s + 2.10e5·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.186·5-s + 0.755·7-s + 9-s − 0.918·11-s − 0.816·13-s + 0.214·15-s + 0.384·17-s + 0.517·19-s − 0.872·21-s + 1.76·23-s − 0.418·25-s − 0.769·27-s + 0.778·29-s − 0.432·31-s + 1.06·33-s − 0.140·35-s − 1.24·37-s + 0.942·39-s − 1.41·41-s − 1.24·43-s − 0.186·45-s + 1.12·47-s + 3/7·49-s − 0.443·51-s − 1.86·53-s + 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 52 T + 7078 p T^{2} + 52 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4056 T + 40048726 T^{2} + 4056 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6468 T + 87341390 T^{2} + 6468 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7780 T + 589699046 T^{2} - 7780 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 15464 T + 1765368966 T^{2} - 15464 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 103184 T + 9202912334 T^{2} - 103184 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 102284 T + 25132293326 T^{2} - 102284 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 71776 T + 46648147902 T^{2} + 71776 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 382404 T + 192539666846 T^{2} + 382404 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 623564 T + 391956280982 T^{2} + 623564 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 647096 T + 636791139414 T^{2} + 647096 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 802880 T + 1172820832670 T^{2} - 802880 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2024068 T + 2689826891006 T^{2} + 2024068 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 546600 T + 4298460506038 T^{2} + 546600 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3751684 T + 8935252937070 T^{2} + 3751684 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 790200 T + 10984305911462 T^{2} - 790200 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1365360 T + 15912981585646 T^{2} + 1365360 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4654924 T + 13685508857622 T^{2} + 4654924 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1208608 T + 38653883863134 T^{2} + 1208608 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9332952 T + 74921955440134 T^{2} + 9332952 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1269068 T + 48938828188214 T^{2} + 1269068 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3805988 T + 165178468073862 T^{2} - 3805988 p^{7} T^{3} + p^{14} T^{4} \) |
| 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
−11.14177899876088242558078909519, −10.87784044262505855262721111523, −10.24650802834325009767713644782, −10.03976822556869484131185099934, −9.189996550034871707256233543631, −8.743403712508364465786836422432, −7.81846579752764238524230719705, −7.72654416255474744795235935817, −6.88955137266692256838141525490, −6.65997450709213499931938559298, −5.47975779244501212481406237831, −5.43369689150916409948082420805, −4.80202485344033596269075263791, −4.36401307053533606162341005527, −3.26074073177202323654958738692, −2.76246540061202464856979901217, −1.60020895052357964642615874436, −1.25621286056939315709073483976, 0, 0,
1.25621286056939315709073483976, 1.60020895052357964642615874436, 2.76246540061202464856979901217, 3.26074073177202323654958738692, 4.36401307053533606162341005527, 4.80202485344033596269075263791, 5.43369689150916409948082420805, 5.47975779244501212481406237831, 6.65997450709213499931938559298, 6.88955137266692256838141525490, 7.72654416255474744795235935817, 7.81846579752764238524230719705, 8.743403712508364465786836422432, 9.189996550034871707256233543631, 10.03976822556869484131185099934, 10.24650802834325009767713644782, 10.87784044262505855262721111523, 11.14177899876088242558078909519
