L(s) = 1 | + 2·5-s − 7-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s − 2·37-s + 2·41-s + 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 8·55-s − 6·61-s + 3·63-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s − 4·77-s + 16·79-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 0.768·61-s + 0.377·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.10392676488786, −12.59818025570201, −12.21061112871525, −11.94096376151410, −11.11149891375802, −10.86987296867177, −10.39177003762592, −9.700651257193109, −9.405201660413936, −9.079731778454922, −8.555996424455293, −7.980269082912205, −7.431576523938057, −6.907496441679553, −6.328830264422105, −5.952583673407794, −5.556023472875278, −5.138418602038677, −4.289788584701189, −3.677973601166209, −3.493445716569914, −2.459486064845408, −2.227939565093547, −1.585166755133977, −0.7762213636292572, 0,
0.7762213636292572, 1.585166755133977, 2.227939565093547, 2.459486064845408, 3.493445716569914, 3.677973601166209, 4.289788584701189, 5.138418602038677, 5.556023472875278, 5.952583673407794, 6.328830264422105, 6.907496441679553, 7.431576523938057, 7.980269082912205, 8.555996424455293, 9.079731778454922, 9.405201660413936, 9.700651257193109, 10.39177003762592, 10.86987296867177, 11.11149891375802, 11.94096376151410, 12.21061112871525, 12.59818025570201, 13.10392676488786