| L(s) = 1 | + 2.53·2-s + 3-s + 4.42·4-s + 1.08·5-s + 2.53·6-s + 1.73·7-s + 6.13·8-s + 9-s + 2.76·10-s + 6.29·11-s + 4.42·12-s − 1.27·13-s + 4.38·14-s + 1.08·15-s + 6.71·16-s − 5.08·17-s + 2.53·18-s − 6.52·19-s + 4.81·20-s + 1.73·21-s + 15.9·22-s − 4.52·23-s + 6.13·24-s − 3.81·25-s − 3.24·26-s + 27-s + 7.65·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.21·4-s + 0.487·5-s + 1.03·6-s + 0.654·7-s + 2.17·8-s + 0.333·9-s + 0.873·10-s + 1.89·11-s + 1.27·12-s − 0.354·13-s + 1.17·14-s + 0.281·15-s + 1.67·16-s − 1.23·17-s + 0.597·18-s − 1.49·19-s + 1.07·20-s + 0.377·21-s + 3.40·22-s − 0.943·23-s + 1.25·24-s − 0.762·25-s − 0.635·26-s + 0.192·27-s + 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.868849964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.868849964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 6.17T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 - 1.83T + 43T^{2} \) |
| 47 | \( 1 - 1.51T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 + 3.08T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.961619376604147569211735462727, −7.962134266162541965796818344275, −7.03884911972320391466864752934, −6.31601287305770577374545311374, −5.88813731519254925399866044326, −4.53003205289997111442189386308, −4.31243359718082797916937918177, −3.47499299024779163638058082960, −2.15217765472860767494570991290, −1.83479030865817921238710486721,
1.83479030865817921238710486721, 2.15217765472860767494570991290, 3.47499299024779163638058082960, 4.31243359718082797916937918177, 4.53003205289997111442189386308, 5.88813731519254925399866044326, 6.31601287305770577374545311374, 7.03884911972320391466864752934, 7.962134266162541965796818344275, 8.961619376604147569211735462727
