| L(s) = 1 | + 0.210·2-s + 0.0798·3-s − 1.95·4-s + 0.0168·6-s + 1.68·7-s − 0.834·8-s − 2.99·9-s − 2.88·11-s − 0.156·12-s − 6.46·13-s + 0.355·14-s + 3.73·16-s − 2.98·17-s − 0.631·18-s + 0.134·21-s − 0.609·22-s − 8.25·23-s − 0.0666·24-s − 1.36·26-s − 0.478·27-s − 3.29·28-s − 7.25·29-s − 4.05·31-s + 2.45·32-s − 0.230·33-s − 0.630·34-s + 5.85·36-s + ⋯ |
| L(s) = 1 | + 0.149·2-s + 0.0461·3-s − 0.977·4-s + 0.00688·6-s + 0.637·7-s − 0.295·8-s − 0.997·9-s − 0.870·11-s − 0.0451·12-s − 1.79·13-s + 0.0951·14-s + 0.933·16-s − 0.724·17-s − 0.148·18-s + 0.0294·21-s − 0.129·22-s − 1.72·23-s − 0.0136·24-s − 0.267·26-s − 0.0921·27-s − 0.623·28-s − 1.34·29-s − 0.728·31-s + 0.434·32-s − 0.0401·33-s − 0.108·34-s + 0.975·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2728231329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2728231329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.210T + 2T^{2} \) |
| 3 | \( 1 - 0.0798T + 3T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 23 | \( 1 + 8.25T + 23T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 + 4.05T + 31T^{2} \) |
| 37 | \( 1 - 7.96T + 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 + 1.64T + 47T^{2} \) |
| 53 | \( 1 - 1.91T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 2.26T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 9.86T + 97T^{2} \) |
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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
−7.87306487802790528606641769838, −7.31140777531845335631958876388, −6.08638346988946203954408887367, −5.57703580867215692883789047890, −4.89943571857794902525733857476, −4.43686306219007412054062804170, −3.51864149501190461725052304965, −2.57996351544509528800610729511, −1.93779796184365924428759314056, −0.23070071225265816930031850902,
0.23070071225265816930031850902, 1.93779796184365924428759314056, 2.57996351544509528800610729511, 3.51864149501190461725052304965, 4.43686306219007412054062804170, 4.89943571857794902525733857476, 5.57703580867215692883789047890, 6.08638346988946203954408887367, 7.31140777531845335631958876388, 7.87306487802790528606641769838
