| L(s) = 1 | + 2.22·2-s + 2.93·4-s + 0.0570·5-s − 3.64·7-s + 2.06·8-s + 0.126·10-s − 3.71·11-s − 0.736·13-s − 8.09·14-s − 1.27·16-s − 1.72·17-s − 0.753·19-s + 0.167·20-s − 8.25·22-s − 1.29·23-s − 4.99·25-s − 1.63·26-s − 10.6·28-s − 2.49·29-s + 6.31·31-s − 6.95·32-s − 3.83·34-s − 0.207·35-s − 1.11·37-s − 1.67·38-s + 0.118·40-s + 0.457·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.46·4-s + 0.0255·5-s − 1.37·7-s + 0.730·8-s + 0.0400·10-s − 1.12·11-s − 0.204·13-s − 2.16·14-s − 0.317·16-s − 0.418·17-s − 0.172·19-s + 0.0374·20-s − 1.75·22-s − 0.270·23-s − 0.999·25-s − 0.320·26-s − 2.01·28-s − 0.462·29-s + 1.13·31-s − 1.22·32-s − 0.657·34-s − 0.0351·35-s − 0.182·37-s − 0.271·38-s + 0.0186·40-s + 0.0715·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 - 0.0570T + 5T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 + 0.736T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 0.753T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 + 2.49T + 29T^{2} \) |
| 31 | \( 1 - 6.31T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 - 0.457T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 - 0.214T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 9.97T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.92T + 83T^{2} \) |
| 89 | \( 1 + 0.440T + 89T^{2} \) |
| 97 | \( 1 + 7.68T + 97T^{2} \) |
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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
−8.672363998801677974289082252860, −7.61287433706771007976653974472, −6.81810783419695244804323275496, −6.09959380843388502794067929871, −5.52603280008222371023049195400, −4.59862996213378650155705500281, −3.78119295870544768717971258865, −2.95968876355688432363099676661, −2.25711911582769317092470742776, 0,
2.25711911582769317092470742776, 2.95968876355688432363099676661, 3.78119295870544768717971258865, 4.59862996213378650155705500281, 5.52603280008222371023049195400, 6.09959380843388502794067929871, 6.81810783419695244804323275496, 7.61287433706771007976653974472, 8.672363998801677974289082252860
