L(s) = 1 | + 5-s + 2.34·7-s − 0.333·11-s + 3.37·13-s − 7.81·17-s + 1.20·19-s + 23-s + 25-s − 1.35·29-s − 10.2·31-s + 2.34·35-s + 2.83·37-s − 7.94·41-s − 7.92·43-s − 11.7·47-s − 1.47·49-s − 7.92·53-s − 0.333·55-s − 13.2·59-s + 2.88·61-s + 3.37·65-s + 7.70·67-s + 15.1·71-s − 7.02·73-s − 0.784·77-s + 0.325·79-s − 10.4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.888·7-s − 0.100·11-s + 0.937·13-s − 1.89·17-s + 0.277·19-s + 0.208·23-s + 0.200·25-s − 0.251·29-s − 1.84·31-s + 0.397·35-s + 0.466·37-s − 1.24·41-s − 1.20·43-s − 1.71·47-s − 0.211·49-s − 1.08·53-s − 0.0450·55-s − 1.72·59-s + 0.369·61-s + 0.419·65-s + 0.941·67-s + 1.80·71-s − 0.821·73-s − 0.0893·77-s + 0.0365·79-s − 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 + 0.333T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 2.88T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 - 0.325T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−7.47666355126852599396980498866, −6.65103439372203577040503644488, −6.22029735915749709846471033980, −5.19764256268268685869102449995, −4.82996407597203751085274565178, −3.90330060302934983104759306787, −3.11006915028375604249485729118, −1.95025456946046942060788146022, −1.55090063688615433536504416990, 0,
1.55090063688615433536504416990, 1.95025456946046942060788146022, 3.11006915028375604249485729118, 3.90330060302934983104759306787, 4.82996407597203751085274565178, 5.19764256268268685869102449995, 6.22029735915749709846471033980, 6.65103439372203577040503644488, 7.47666355126852599396980498866