L(s) = 1 | − 5-s + 2.37·7-s − 11-s + 2·13-s + 4.37·17-s − 6.37·19-s − 8.74·23-s + 25-s + 4.37·29-s + 2.37·31-s − 2.37·35-s + 3.62·37-s − 11.4·41-s + 4·43-s − 8.74·47-s − 1.37·49-s − 13.1·53-s + 55-s + 8.74·59-s + 0.372·61-s − 2·65-s − 8·67-s − 7.11·71-s + 7.48·73-s − 2.37·77-s + 12.7·79-s + 8.74·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.896·7-s − 0.301·11-s + 0.554·13-s + 1.06·17-s − 1.46·19-s − 1.82·23-s + 0.200·25-s + 0.811·29-s + 0.426·31-s − 0.400·35-s + 0.596·37-s − 1.79·41-s + 0.609·43-s − 1.27·47-s − 0.196·49-s − 1.80·53-s + 0.134·55-s + 1.13·59-s + 0.0476·61-s − 0.248·65-s − 0.977·67-s − 0.844·71-s + 0.876·73-s − 0.270·77-s + 1.43·79-s + 0.959·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−7.900047705474602046337235309368, −6.68771281532169450009539440474, −6.21226239667441634392949680735, −5.33246190741386796214907863313, −4.62017706240319685030803708526, −3.99173928422712151225574323017, −3.17919496497643753078867498300, −2.13342393124760178099424414401, −1.33598046409242204272016219604, 0,
1.33598046409242204272016219604, 2.13342393124760178099424414401, 3.17919496497643753078867498300, 3.99173928422712151225574323017, 4.62017706240319685030803708526, 5.33246190741386796214907863313, 6.21226239667441634392949680735, 6.68771281532169450009539440474, 7.900047705474602046337235309368