| L(s) = 1 | + 2.30·2-s − 0.0652·3-s + 3.31·4-s + 2.38·5-s − 0.150·6-s + 2.55·7-s + 3.03·8-s − 2.99·9-s + 5.50·10-s − 2.99·11-s − 0.216·12-s + 3.21·13-s + 5.88·14-s − 0.155·15-s + 0.358·16-s + 17-s − 6.90·18-s + 1.77·19-s + 7.91·20-s − 0.166·21-s − 6.90·22-s − 8.18·23-s − 0.197·24-s + 0.703·25-s + 7.42·26-s + 0.391·27-s + 8.46·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.0376·3-s + 1.65·4-s + 1.06·5-s − 0.0614·6-s + 0.965·7-s + 1.07·8-s − 0.998·9-s + 1.74·10-s − 0.903·11-s − 0.0624·12-s + 0.892·13-s + 1.57·14-s − 0.0402·15-s + 0.0896·16-s + 0.242·17-s − 1.62·18-s + 0.406·19-s + 1.77·20-s − 0.0363·21-s − 1.47·22-s − 1.70·23-s − 0.0403·24-s + 0.140·25-s + 1.45·26-s + 0.0753·27-s + 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.247544757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.247544757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.0652T + 3T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 - 2.36T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 6.73T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.72340138663528638960759056584, −9.726177823685431709400046979307, −8.499487932269461318684965528257, −7.69309062718194662719432647223, −6.24266967865938328565210134903, −5.74896477496353686712753641382, −5.14480193242845092350961540394, −4.03885593022202002477056833714, −2.82596517862934764424211265981, −1.90706477126586889456208308937,
1.90706477126586889456208308937, 2.82596517862934764424211265981, 4.03885593022202002477056833714, 5.14480193242845092350961540394, 5.74896477496353686712753641382, 6.24266967865938328565210134903, 7.69309062718194662719432647223, 8.499487932269461318684965528257, 9.726177823685431709400046979307, 10.72340138663528638960759056584
