| L(s) = 1 | − 2.52·2-s + 2.20·3-s + 4.35·4-s − 5.54·6-s + 1.38·7-s − 5.95·8-s + 1.84·9-s + 1.13·11-s + 9.59·12-s + 1.42·13-s − 3.49·14-s + 6.28·16-s − 8.05·17-s − 4.64·18-s + 8.18·19-s + 3.04·21-s − 2.87·22-s − 6.39·23-s − 13.0·24-s − 3.59·26-s − 2.55·27-s + 6.03·28-s − 1.29·29-s − 5.90·31-s − 3.95·32-s + 2.50·33-s + 20.3·34-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 1.27·3-s + 2.17·4-s − 2.26·6-s + 0.523·7-s − 2.10·8-s + 0.613·9-s + 0.343·11-s + 2.76·12-s + 0.395·13-s − 0.933·14-s + 1.57·16-s − 1.95·17-s − 1.09·18-s + 1.87·19-s + 0.664·21-s − 0.612·22-s − 1.33·23-s − 2.67·24-s − 0.705·26-s − 0.490·27-s + 1.14·28-s − 0.239·29-s − 1.06·31-s − 0.699·32-s + 0.436·33-s + 3.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 - 2.20T + 3T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 8.05T + 17T^{2} \) |
| 19 | \( 1 - 8.18T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 - 3.02T + 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 2.05T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.056163845860444785098565052143, −7.40729881546258325591787641501, −6.72855496945958068729446482754, −5.94476712980073118038934380133, −4.73046545743630377016369200342, −3.67890141803194153421559235947, −2.88129504427923715263666194313, −1.97309103278989928094662225246, −1.49405569376506580207009281936, 0,
1.49405569376506580207009281936, 1.97309103278989928094662225246, 2.88129504427923715263666194313, 3.67890141803194153421559235947, 4.73046545743630377016369200342, 5.94476712980073118038934380133, 6.72855496945958068729446482754, 7.40729881546258325591787641501, 8.056163845860444785098565052143
