| L(s) = 1 | − 38.1·2-s − 47.8·3-s + 944.·4-s + 1.82e3·6-s − 5.94e3·7-s − 1.65e4·8-s − 1.73e4·9-s − 2.52e4·11-s − 4.52e4·12-s − 2.85e4·13-s + 2.27e5·14-s + 1.46e5·16-s − 1.09e5·17-s + 6.63e5·18-s − 9.04e5·19-s + 2.84e5·21-s + 9.62e5·22-s + 4.35e5·23-s + 7.90e5·24-s + 1.09e6·26-s + 1.77e6·27-s − 5.61e6·28-s + 6.44e6·29-s + 6.62e6·31-s + 2.85e6·32-s + 1.20e6·33-s + 4.17e6·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.341·3-s + 1.84·4-s + 0.575·6-s − 0.936·7-s − 1.42·8-s − 0.883·9-s − 0.519·11-s − 0.629·12-s − 0.277·13-s + 1.57·14-s + 0.560·16-s − 0.317·17-s + 1.49·18-s − 1.59·19-s + 0.319·21-s + 0.875·22-s + 0.324·23-s + 0.486·24-s + 0.467·26-s + 0.642·27-s − 1.72·28-s + 1.69·29-s + 1.28·31-s + 0.481·32-s + 0.177·33-s + 0.535·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 38.1T + 512T^{2} \) |
| 3 | \( 1 + 47.8T + 1.96e4T^{2} \) |
| 7 | \( 1 + 5.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.52e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.49e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.30e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.52e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.34e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.47e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.25e9T + 7.60e17T^{2} \) |
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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
−9.616923308190313599127138841411, −8.591815666833785570449583562481, −8.108650669460197208446540433912, −6.68815529970307049944807014989, −6.32806225837864334263139098698, −4.79719901880306042619058504192, −3.02935489693001286979332919253, −2.19857658429334181861044127635, −0.71696929860210428696086758387, 0,
0.71696929860210428696086758387, 2.19857658429334181861044127635, 3.02935489693001286979332919253, 4.79719901880306042619058504192, 6.32806225837864334263139098698, 6.68815529970307049944807014989, 8.108650669460197208446540433912, 8.591815666833785570449583562481, 9.616923308190313599127138841411
