| L(s) = 1 | + 1.05·2-s − 126.·4-s − 151.·5-s − 1.57e3·7-s − 269.·8-s − 160.·10-s − 1.59e3·11-s + 8.75e3·13-s − 1.66e3·14-s + 1.59e4·16-s − 1.98e4·17-s − 4.83e3·19-s + 1.92e4·20-s − 1.69e3·22-s + 3.07e4·23-s − 5.51e4·25-s + 9.27e3·26-s + 1.99e5·28-s − 1.47e4·29-s + 7.22e4·31-s + 5.14e4·32-s − 2.10e4·34-s + 2.38e5·35-s + 3.57e5·37-s − 5.12e3·38-s + 4.09e4·40-s − 5.62e5·41-s + ⋯ |
| L(s) = 1 | + 0.0935·2-s − 0.991·4-s − 0.542·5-s − 1.73·7-s − 0.186·8-s − 0.0507·10-s − 0.361·11-s + 1.10·13-s − 0.162·14-s + 0.973·16-s − 0.979·17-s − 0.161·19-s + 0.537·20-s − 0.0338·22-s + 0.526·23-s − 0.705·25-s + 0.103·26-s + 1.71·28-s − 0.112·29-s + 0.435·31-s + 0.277·32-s − 0.0916·34-s + 0.941·35-s + 1.16·37-s − 0.0151·38-s + 0.101·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 1.05T + 128T^{2} \) |
| 5 | \( 1 + 151.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.57e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.75e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.98e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.83e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.07e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.47e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.22e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.57e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.82e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.94e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.48e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.02e7T + 8.07e13T^{2} \) |
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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
−9.160283984361927803662747317500, −8.617807117216679711803126933574, −7.48239610049309386459243170932, −6.40322442420145390131622483063, −5.66339819544405808768660669428, −4.28123234428584336097186771978, −3.68946101238816043880113561595, −2.68760140436763411480205537361, −0.818915492883988648642269893901, 0,
0.818915492883988648642269893901, 2.68760140436763411480205537361, 3.68946101238816043880113561595, 4.28123234428584336097186771978, 5.66339819544405808768660669428, 6.40322442420145390131622483063, 7.48239610049309386459243170932, 8.617807117216679711803126933574, 9.160283984361927803662747317500
