L(s) = 1 | − 2-s + 4-s − 5-s + 1.07·7-s − 8-s + 10-s − 6.04·11-s − 0.290·13-s − 1.07·14-s + 16-s − 1.07·17-s + 5.26·19-s − 20-s + 6.04·22-s + 4.34·23-s + 25-s + 0.290·26-s + 1.07·28-s + 9.31·29-s + 31-s − 32-s + 1.07·34-s − 1.07·35-s − 2.44·37-s − 5.26·38-s + 40-s − 5.60·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.407·7-s − 0.353·8-s + 0.316·10-s − 1.82·11-s − 0.0806·13-s − 0.288·14-s + 0.250·16-s − 0.261·17-s + 1.20·19-s − 0.223·20-s + 1.28·22-s + 0.904·23-s + 0.200·25-s + 0.0570·26-s + 0.203·28-s + 1.72·29-s + 0.179·31-s − 0.176·32-s + 0.184·34-s − 0.182·35-s − 0.402·37-s − 0.853·38-s + 0.158·40-s − 0.874·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 + 0.290T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 9.31T + 29T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 8.18T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 3.84T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−8.352588274085450218443839301328, −7.75610669420067245234949282626, −7.21529294736749966187554232683, −6.27818910681276494194686223980, −5.14889988964788260392122932379, −4.77203662143397282599381711181, −3.24609115963682442183669842703, −2.66437745415036059099628702332, −1.33947502059879646237763624997, 0,
1.33947502059879646237763624997, 2.66437745415036059099628702332, 3.24609115963682442183669842703, 4.77203662143397282599381711181, 5.14889988964788260392122932379, 6.27818910681276494194686223980, 7.21529294736749966187554232683, 7.75610669420067245234949282626, 8.352588274085450218443839301328