| L(s) = 1 | + 16·2-s + 256·4-s + 76·5-s − 2.40e3·7-s + 4.09e3·8-s + 1.21e3·10-s − 3.83e4·11-s + 9.82e4·13-s − 3.84e4·14-s + 6.55e4·16-s − 1.04e5·17-s − 4.20e5·19-s + 1.94e4·20-s − 6.14e5·22-s + 1.39e5·23-s − 1.94e6·25-s + 1.57e6·26-s − 6.14e5·28-s + 1.91e6·29-s − 6.37e6·31-s + 1.04e6·32-s − 1.67e6·34-s − 1.82e5·35-s − 6.62e6·37-s − 6.72e6·38-s + 3.11e5·40-s + 6.69e6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.0543·5-s − 0.377·7-s + 0.353·8-s + 0.0384·10-s − 0.790·11-s + 0.954·13-s − 0.267·14-s + 1/4·16-s − 0.303·17-s − 0.740·19-s + 0.0271·20-s − 0.558·22-s + 0.103·23-s − 0.997·25-s + 0.674·26-s − 0.188·28-s + 0.503·29-s − 1.24·31-s + 0.176·32-s − 0.214·34-s − 0.0205·35-s − 0.581·37-s − 0.523·38-s + 0.0192·40-s + 0.369·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{4} T \) |
| good | 5 | \( 1 - 76 T + p^{9} T^{2} \) |
| 11 | \( 1 + 38386 T + p^{9} T^{2} \) |
| 13 | \( 1 - 98298 T + p^{9} T^{2} \) |
| 17 | \( 1 + 104524 T + p^{9} T^{2} \) |
| 19 | \( 1 + 420580 T + p^{9} T^{2} \) |
| 23 | \( 1 - 139118 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1916290 T + p^{9} T^{2} \) |
| 31 | \( 1 + 6379488 T + p^{9} T^{2} \) |
| 37 | \( 1 + 6629278 T + p^{9} T^{2} \) |
| 41 | \( 1 - 6692112 T + p^{9} T^{2} \) |
| 43 | \( 1 + 23269732 T + p^{9} T^{2} \) |
| 47 | \( 1 - 22000596 T + p^{9} T^{2} \) |
| 53 | \( 1 + 18919770 T + p^{9} T^{2} \) |
| 59 | \( 1 + 179035544 T + p^{9} T^{2} \) |
| 61 | \( 1 + 19797786 T + p^{9} T^{2} \) |
| 67 | \( 1 + 263015240 T + p^{9} T^{2} \) |
| 71 | \( 1 + 22447678 T + p^{9} T^{2} \) |
| 73 | \( 1 - 11023774 T + p^{9} T^{2} \) |
| 79 | \( 1 + 284917908 T + p^{9} T^{2} \) |
| 83 | \( 1 - 226865924 T + p^{9} T^{2} \) |
| 89 | \( 1 - 191377296 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1162236578 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.12548880433953233165858242577, −10.32875349853610564709375075517, −8.952016818320288380594342998172, −7.74773449467280672173043032146, −6.49653748789501218691870146723, −5.55255787380191511049042235538, −4.23590509170703146104065890782, −3.08028349299865785556327885805, −1.75478836261226220616416623615, 0,
1.75478836261226220616416623615, 3.08028349299865785556327885805, 4.23590509170703146104065890782, 5.55255787380191511049042235538, 6.49653748789501218691870146723, 7.74773449467280672173043032146, 8.952016818320288380594342998172, 10.32875349853610564709375075517, 11.12548880433953233165858242577
