| L(s) = 1 | + 2-s + 1.40·3-s + 4-s − 0.425·5-s + 1.40·6-s + 2.92·7-s + 8-s − 1.03·9-s − 0.425·10-s − 6.25·11-s + 1.40·12-s + 2.92·14-s − 0.597·15-s + 16-s − 4.36·17-s − 1.03·18-s − 19-s − 0.425·20-s + 4.10·21-s − 6.25·22-s + 2.24·23-s + 1.40·24-s − 4.81·25-s − 5.65·27-s + 2.92·28-s + 0.394·29-s − 0.597·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.810·3-s + 0.5·4-s − 0.190·5-s + 0.572·6-s + 1.10·7-s + 0.353·8-s − 0.343·9-s − 0.134·10-s − 1.88·11-s + 0.405·12-s + 0.781·14-s − 0.154·15-s + 0.250·16-s − 1.05·17-s − 0.242·18-s − 0.229·19-s − 0.0952·20-s + 0.895·21-s − 1.33·22-s + 0.468·23-s + 0.286·24-s − 0.963·25-s − 1.08·27-s + 0.552·28-s + 0.0732·29-s − 0.109·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 + 0.425T + 5T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 + 6.25T + 11T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 + 4.60T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 8.23T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 7.39T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 + 0.552T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
| 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
−7.69160848162270859680561977862, −7.19695881476006013181069504948, −6.08924929760739756846197882584, −5.30535775466995067737827683643, −4.87017105708213265790400948454, −4.00432894816695186418070138307, −3.14603560069952093641435989950, −2.38238052640107643069560060434, −1.83737432281091743768684586185, 0,
1.83737432281091743768684586185, 2.38238052640107643069560060434, 3.14603560069952093641435989950, 4.00432894816695186418070138307, 4.87017105708213265790400948454, 5.30535775466995067737827683643, 6.08924929760739756846197882584, 7.19695881476006013181069504948, 7.69160848162270859680561977862
