| L(s) = 1 | + 2-s + 4-s − 1.30·5-s + 3.60·7-s + 8-s − 1.30·10-s − 3.30·13-s + 3.60·14-s + 16-s + 2.60·17-s − 3.69·19-s − 1.30·20-s − 6.90·23-s − 3.30·25-s − 3.30·26-s + 3.60·28-s − 0.394·29-s − 7.51·31-s + 32-s + 2.60·34-s − 4.69·35-s − 3.60·37-s − 3.69·38-s − 1.30·40-s + 0.394·41-s − 10.2·43-s − 6.90·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.582·5-s + 1.36·7-s + 0.353·8-s − 0.411·10-s − 0.916·13-s + 0.963·14-s + 0.250·16-s + 0.631·17-s − 0.848·19-s − 0.291·20-s − 1.44·23-s − 0.660·25-s − 0.647·26-s + 0.681·28-s − 0.0732·29-s − 1.34·31-s + 0.176·32-s + 0.446·34-s − 0.793·35-s − 0.592·37-s − 0.599·38-s − 0.205·40-s + 0.0616·41-s − 1.55·43-s − 1.01·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 0.394T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 0.394T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 9.51T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.78613648334571574257171042612, −7.01980098898798220741975071259, −6.12032047799997094545502804898, −5.33005958902855868042518099150, −4.78997191685902225263823654523, −4.06992697653204936319973296723, −3.42397014043437481619008180271, −2.19490176359963424619297085583, −1.66140241696923516260188256890, 0,
1.66140241696923516260188256890, 2.19490176359963424619297085583, 3.42397014043437481619008180271, 4.06992697653204936319973296723, 4.78997191685902225263823654523, 5.33005958902855868042518099150, 6.12032047799997094545502804898, 7.01980098898798220741975071259, 7.78613648334571574257171042612
