| L(s) = 1 | + 2.30·2-s + 2.89·3-s + 3.30·4-s + 6.65·6-s − 3.91·7-s + 2.99·8-s + 5.35·9-s + 2.65·11-s + 9.53·12-s − 5.62·13-s − 9.00·14-s + 0.292·16-s + 1.86·17-s + 12.3·18-s + 1.69·19-s − 11.3·21-s + 6.11·22-s − 0.691·23-s + 8.65·24-s − 12.9·26-s + 6.80·27-s − 12.9·28-s − 29-s − 0.654·31-s − 5.31·32-s + 7.67·33-s + 4.30·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.66·3-s + 1.65·4-s + 2.71·6-s − 1.47·7-s + 1.05·8-s + 1.78·9-s + 0.800·11-s + 2.75·12-s − 1.56·13-s − 2.40·14-s + 0.0731·16-s + 0.453·17-s + 2.90·18-s + 0.389·19-s − 2.46·21-s + 1.30·22-s − 0.144·23-s + 1.76·24-s − 2.54·26-s + 1.30·27-s − 2.44·28-s − 0.185·29-s − 0.117·31-s − 0.939·32-s + 1.33·33-s + 0.737·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.288900298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.288900298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 + 3.91T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 5.62T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 + 0.691T + 23T^{2} \) |
| 31 | \( 1 + 0.654T + 31T^{2} \) |
| 37 | \( 1 - 3.91T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 - 3.30T + 89T^{2} \) |
| 97 | \( 1 - 0.384T + 97T^{2} \) |
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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
−10.11605404505447719950764634853, −9.595719689301144789469819221549, −8.788140331612134343496437173106, −7.43521885957918433016019969711, −6.91917217921321185199441357187, −5.87097896980374721034917488230, −4.61162165056422759931215073924, −3.65030487628230273612626578869, −3.08633944712548318569763694450, −2.22641937343683532079674989496,
2.22641937343683532079674989496, 3.08633944712548318569763694450, 3.65030487628230273612626578869, 4.61162165056422759931215073924, 5.87097896980374721034917488230, 6.91917217921321185199441357187, 7.43521885957918433016019969711, 8.788140331612134343496437173106, 9.595719689301144789469819221549, 10.11605404505447719950764634853
