| L(s) = 1 | + 1.83·2-s + 3-s + 1.35·4-s + 5-s + 1.83·6-s − 1.47·7-s − 1.17·8-s + 9-s + 1.83·10-s − 1.03·11-s + 1.35·12-s + 2.31·13-s − 2.71·14-s + 15-s − 4.87·16-s − 6.53·17-s + 1.83·18-s − 2.23·19-s + 1.35·20-s − 1.47·21-s − 1.89·22-s − 1.17·24-s + 25-s + 4.23·26-s + 27-s − 2.00·28-s − 0.725·29-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.577·3-s + 0.678·4-s + 0.447·5-s + 0.748·6-s − 0.559·7-s − 0.416·8-s + 0.333·9-s + 0.579·10-s − 0.311·11-s + 0.391·12-s + 0.641·13-s − 0.724·14-s + 0.258·15-s − 1.21·16-s − 1.58·17-s + 0.431·18-s − 0.513·19-s + 0.303·20-s − 0.322·21-s − 0.404·22-s − 0.240·24-s + 0.200·25-s + 0.830·26-s + 0.192·27-s − 0.379·28-s − 0.134·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 29 | \( 1 + 0.725T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 5.34T + 47T^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 1.05T + 61T^{2} \) |
| 67 | \( 1 + 2.28T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 + 5.74T + 79T^{2} \) |
| 83 | \( 1 + 3.13T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.27647745059524736284157639431, −6.47334307703313405378614589001, −6.15410667085462306458885877771, −5.34054821313108936546676022784, −4.46791709118167594868290868508, −4.05557986008278231273439498150, −3.12824367012157628260917790289, −2.58788946138009423763632618378, −1.70779569345026947760004560076, 0,
1.70779569345026947760004560076, 2.58788946138009423763632618378, 3.12824367012157628260917790289, 4.05557986008278231273439498150, 4.46791709118167594868290868508, 5.34054821313108936546676022784, 6.15410667085462306458885877771, 6.47334307703313405378614589001, 7.27647745059524736284157639431
