| L(s) = 1 | + 2-s − 7·4-s − 10·5-s + 9·7-s − 7·8-s − 10·10-s + 37·11-s − 112·13-s + 9·14-s − 7·16-s − 77·17-s + 35·19-s + 70·20-s + 37·22-s + 267·23-s + 75·25-s − 112·26-s − 63·28-s − 325·29-s − 12·31-s − 71·32-s − 77·34-s − 90·35-s − 638·37-s + 35·38-s + 70·40-s − 238·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 0.894·5-s + 0.485·7-s − 0.309·8-s − 0.316·10-s + 1.01·11-s − 2.38·13-s + 0.171·14-s − 0.109·16-s − 1.09·17-s + 0.422·19-s + 0.782·20-s + 0.358·22-s + 2.42·23-s + 3/5·25-s − 0.844·26-s − 0.425·28-s − 2.08·29-s − 0.0695·31-s − 0.392·32-s − 0.388·34-s − 0.434·35-s − 2.83·37-s + 0.149·38-s + 0.276·40-s − 0.906·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 698 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 37 T + 2798 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 112 T + 7002 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 77 T + 6152 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 35 T + 8868 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 267 T + 37792 T^{2} - 267 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 325 T + 62636 T^{2} + 325 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 638 T + 5466 p T^{2} + 638 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 238 T + 2983 p T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 97 T + 98922 T^{2} + 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 901 T + 409598 T^{2} - 901 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 224 T + 227798 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 85 T + 392756 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 247 T + 155508 T^{2} + 247 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 606 T + 414023 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 394 T + 654806 T^{2} + 394 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 811 T + 595758 T^{2} + 811 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 840 T + 1114826 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 387 T - 511958 T^{2} - 387 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1065 T + 874888 T^{2} - 1065 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1031 T + 1949508 T^{2} + 1031 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−10.71667543966749900782659972660, −10.18177171908225922826524712617, −9.431624150473441886638381630360, −9.264583893595391857285401077851, −8.709502581352970438727421516140, −8.641444849449790634356388908158, −7.55055389687506104205181752667, −7.21903339730119071028053050655, −7.17254545231116624289259380952, −6.47223460530932623288945775323, −5.28060038760387474775933088375, −5.27567504975223743652264756551, −4.60444748632254013808338943803, −4.36734669180147430087365353080, −3.58467266697199202537324885643, −3.09630354773797817563931502604, −2.17241404499691055524953048732, −1.38462346633516835618027143510, 0, 0,
1.38462346633516835618027143510, 2.17241404499691055524953048732, 3.09630354773797817563931502604, 3.58467266697199202537324885643, 4.36734669180147430087365353080, 4.60444748632254013808338943803, 5.27567504975223743652264756551, 5.28060038760387474775933088375, 6.47223460530932623288945775323, 7.17254545231116624289259380952, 7.21903339730119071028053050655, 7.55055389687506104205181752667, 8.641444849449790634356388908158, 8.709502581352970438727421516140, 9.264583893595391857285401077851, 9.431624150473441886638381630360, 10.18177171908225922826524712617, 10.71667543966749900782659972660
