| L(s) = 1 | − 30.3·2-s + 124.·3-s + 410.·4-s − 3.78e3·6-s + 2.69e3·7-s + 3.07e3·8-s − 4.12e3·9-s − 7.58e4·11-s + 5.12e4·12-s + 2.85e4·13-s − 8.19e4·14-s − 3.03e5·16-s + 5.39e5·17-s + 1.25e5·18-s − 4.52e5·19-s + 3.36e5·21-s + 2.30e6·22-s + 3.33e5·23-s + 3.83e5·24-s − 8.67e5·26-s − 2.96e6·27-s + 1.10e6·28-s + 5.02e6·29-s + 6.26e3·31-s + 7.65e6·32-s − 9.45e6·33-s − 1.63e7·34-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.888·3-s + 0.802·4-s − 1.19·6-s + 0.424·7-s + 0.265·8-s − 0.209·9-s − 1.56·11-s + 0.713·12-s + 0.277·13-s − 0.570·14-s − 1.15·16-s + 1.56·17-s + 0.281·18-s − 0.796·19-s + 0.377·21-s + 2.09·22-s + 0.248·23-s + 0.235·24-s − 0.372·26-s − 1.07·27-s + 0.340·28-s + 1.31·29-s + 0.00121·31-s + 1.29·32-s − 1.38·33-s − 2.10·34-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 | \( 1 + 30.3T + 512T^{2} \) |
| 3 | \( 1 - 124.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 2.69e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.58e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 5.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.52e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.26e3T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.08e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.67e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.59e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.09e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.98e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.39e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.48e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.67e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.79e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.54e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.30e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.64e8T + 7.60e17T^{2} \) |
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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
−9.440806432868075696155399556896, −8.508234549994629872794937101309, −8.020339142001990681488369306877, −7.38826333822545621115483231522, −5.82681149939431216258700501882, −4.62664491181312314143871815177, −3.11104099976501640789833949580, −2.25847138478916490495368649758, −1.11575841226070070193518045610, 0,
1.11575841226070070193518045610, 2.25847138478916490495368649758, 3.11104099976501640789833949580, 4.62664491181312314143871815177, 5.82681149939431216258700501882, 7.38826333822545621115483231522, 8.020339142001990681488369306877, 8.508234549994629872794937101309, 9.440806432868075696155399556896
