| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 4·7-s + 9-s + 11-s + 2·12-s − 13-s − 8·14-s − 4·16-s + 7·17-s + 2·18-s − 4·21-s + 2·22-s + 4·23-s − 5·25-s − 2·26-s + 27-s − 8·28-s − 8·29-s + 2·31-s − 8·32-s + 33-s + 14·34-s + 2·36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 0.277·13-s − 2.13·14-s − 16-s + 1.69·17-s + 0.471·18-s − 0.872·21-s + 0.426·22-s + 0.834·23-s − 25-s − 0.392·26-s + 0.192·27-s − 1.51·28-s − 1.48·29-s + 0.359·31-s − 1.41·32-s + 0.174·33-s + 2.40·34-s + 1/3·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.30861880565620, −16.05343337190916, −15.21717159258462, −14.97860074332654, −14.30521348274289, −13.76055386245281, −13.33157645407716, −12.80940093416246, −12.20297564186370, −12.00189271047861, −11.03506617471207, −10.26520067404553, −9.603725554732882, −9.274961531992437, −8.522823710708174, −7.396186232741232, −7.214550838092229, −6.283591096541134, −5.738521553646392, −5.276302545029538, −4.130496739094447, −3.801473852170079, −3.048035862948643, −2.695043274381687, −1.466989757197803, 0,
1.466989757197803, 2.695043274381687, 3.048035862948643, 3.801473852170079, 4.130496739094447, 5.276302545029538, 5.738521553646392, 6.283591096541134, 7.214550838092229, 7.396186232741232, 8.522823710708174, 9.274961531992437, 9.603725554732882, 10.26520067404553, 11.03506617471207, 12.00189271047861, 12.20297564186370, 12.80940093416246, 13.33157645407716, 13.76055386245281, 14.30521348274289, 14.97860074332654, 15.21717159258462, 16.05343337190916, 16.30861880565620
