| L(s) = 1 | − 2.51·2-s − 3-s + 4.34·4-s + 2.51·6-s − 0.173·7-s − 5.90·8-s + 9-s + 5.08·11-s − 4.34·12-s − 1.82·13-s + 0.436·14-s + 6.19·16-s + 4.24·17-s − 2.51·18-s − 8.62·19-s + 0.173·21-s − 12.8·22-s − 3.16·23-s + 5.90·24-s + 4.60·26-s − 27-s − 0.753·28-s + 29-s + 3.22·31-s − 3.78·32-s − 5.08·33-s − 10.6·34-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.17·4-s + 1.02·6-s − 0.0655·7-s − 2.08·8-s + 0.333·9-s + 1.53·11-s − 1.25·12-s − 0.506·13-s + 0.116·14-s + 1.54·16-s + 1.02·17-s − 0.593·18-s − 1.97·19-s + 0.0378·21-s − 2.72·22-s − 0.659·23-s + 1.20·24-s + 0.902·26-s − 0.192·27-s − 0.142·28-s + 0.185·29-s + 0.578·31-s − 0.669·32-s − 0.884·33-s − 1.83·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 7 | \( 1 + 0.173T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 8.62T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 31 | \( 1 - 3.22T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 - 4.54T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−8.634289428123115981019371274873, −8.188114274881213956221648721078, −7.14982176866421071439925990887, −6.57881900002486994031333451663, −6.01355311778597859270131890359, −4.67338120914734568974097030334, −3.56343859013510770513136402419, −2.14254364341070777381892870074, −1.28165043048159175423304649322, 0,
1.28165043048159175423304649322, 2.14254364341070777381892870074, 3.56343859013510770513136402419, 4.67338120914734568974097030334, 6.01355311778597859270131890359, 6.57881900002486994031333451663, 7.14982176866421071439925990887, 8.188114274881213956221648721078, 8.634289428123115981019371274873
