| L(s) = 1 | + 17.9·2-s − 27·3-s + 193.·4-s + 1.22·5-s − 484.·6-s − 719.·7-s + 1.18e3·8-s + 729·9-s + 21.9·10-s + 1.11e3·11-s − 5.23e3·12-s + 8.28e3·13-s − 1.29e4·14-s − 33.0·15-s − 3.58e3·16-s + 4.83e3·17-s + 1.30e4·18-s − 3.16e4·19-s + 237.·20-s + 1.94e4·21-s + 1.99e4·22-s − 4.35e4·23-s − 3.19e4·24-s − 7.81e4·25-s + 1.48e5·26-s − 1.96e4·27-s − 1.39e5·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.51·4-s + 0.00437·5-s − 0.915·6-s − 0.792·7-s + 0.817·8-s + 0.333·9-s + 0.00694·10-s + 0.251·11-s − 0.875·12-s + 1.04·13-s − 1.25·14-s − 0.00252·15-s − 0.218·16-s + 0.238·17-s + 0.528·18-s − 1.05·19-s + 0.00663·20-s + 0.457·21-s + 0.399·22-s − 0.746·23-s − 0.472·24-s − 0.999·25-s + 1.65·26-s − 0.192·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 17.9T + 128T^{2} \) |
| 5 | \( 1 - 1.22T + 7.81e4T^{2} \) |
| 7 | \( 1 + 719.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.11e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.28e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.83e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 8.53e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.27e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.53e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.25e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.29e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.14e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.36e7T + 8.07e13T^{2} \) |
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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
−11.29681396378469317044961330783, −10.25343450965525419423317687242, −8.913921767428343817418970609080, −7.24358436474499742477705010155, −6.14239498243104375484921474110, −5.68205034291030480828999791834, −4.19435473682409357227715477557, −3.50933778246288196030918229703, −1.92961638747251886527349150656, 0,
1.92961638747251886527349150656, 3.50933778246288196030918229703, 4.19435473682409357227715477557, 5.68205034291030480828999791834, 6.14239498243104375484921474110, 7.24358436474499742477705010155, 8.913921767428343817418970609080, 10.25343450965525419423317687242, 11.29681396378469317044961330783
