| L(s) = 1 | + 3.82·2-s + 6.65·4-s − 15.3·5-s − 16.8·7-s − 5.14·8-s − 58.6·10-s + 55.7·11-s − 46.9·13-s − 64.4·14-s − 72.9·16-s − 62.1·17-s − 141.·19-s − 101.·20-s + 213.·22-s − 23·23-s + 109.·25-s − 179.·26-s − 112.·28-s + 288.·29-s + 68.7·31-s − 238.·32-s − 237.·34-s + 257.·35-s + 179.·37-s − 543.·38-s + 78.7·40-s + 71.5·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.832·4-s − 1.36·5-s − 0.908·7-s − 0.227·8-s − 1.85·10-s + 1.52·11-s − 1.00·13-s − 1.22·14-s − 1.13·16-s − 0.886·17-s − 1.71·19-s − 1.13·20-s + 2.07·22-s − 0.208·23-s + 0.876·25-s − 1.35·26-s − 0.756·28-s + 1.84·29-s + 0.398·31-s − 1.31·32-s − 1.20·34-s + 1.24·35-s + 0.798·37-s − 2.31·38-s + 0.311·40-s + 0.272·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 7 | \( 1 + 16.8T + 343T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 71.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 159.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 426.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 81.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 696.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 719.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−12.06132786428238794874788048828, −10.89779754449175077336215185777, −9.430446786063440349718171838769, −8.394959704738237188382798298039, −6.85113279139932214816535044586, −6.32788804697468396728078837971, −4.49051048686462884503053360268, −4.06630158404476451562751306623, −2.79370264610506379733462122524, 0,
2.79370264610506379733462122524, 4.06630158404476451562751306623, 4.49051048686462884503053360268, 6.32788804697468396728078837971, 6.85113279139932214816535044586, 8.394959704738237188382798298039, 9.430446786063440349718171838769, 10.89779754449175077336215185777, 12.06132786428238794874788048828
