L(s) = 1 | + 2.61·2-s + 3-s + 4.81·4-s − 5-s + 2.61·6-s + 4.73·7-s + 7.35·8-s + 9-s − 2.61·10-s + 3.07·11-s + 4.81·12-s − 6.05·13-s + 12.3·14-s − 15-s + 9.57·16-s + 6.93·17-s + 2.61·18-s + 1.19·19-s − 4.81·20-s + 4.73·21-s + 8.02·22-s + 7.35·24-s + 25-s − 15.8·26-s + 27-s + 22.8·28-s − 4.71·29-s + ⋯ |
L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.40·4-s − 0.447·5-s + 1.06·6-s + 1.79·7-s + 2.60·8-s + 0.333·9-s − 0.825·10-s + 0.927·11-s + 1.39·12-s − 1.68·13-s + 3.30·14-s − 0.258·15-s + 2.39·16-s + 1.68·17-s + 0.615·18-s + 0.273·19-s − 1.07·20-s + 1.03·21-s + 1.71·22-s + 1.50·24-s + 0.200·25-s − 3.10·26-s + 0.192·27-s + 4.31·28-s − 0.874·29-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
\(L(1)\) |
\(\approx\) |
\(10.57224259\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.57224259\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.61T + 2T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 + 9.32T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 3.93T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 - 2.18T + 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.76771279224535161954925552598, −7.15182259887483695461833666199, −6.34985515356361131796655910575, −5.33369031247177956944733370001, −4.90447080852105022544389551920, −4.46324327856574642333930251661, −3.56267048977505257197824127341, −3.02533416708230255208023660064, −1.97445779308897079730633336811, −1.41616370440354208248751698663,
1.41616370440354208248751698663, 1.97445779308897079730633336811, 3.02533416708230255208023660064, 3.56267048977505257197824127341, 4.46324327856574642333930251661, 4.90447080852105022544389551920, 5.33369031247177956944733370001, 6.34985515356361131796655910575, 7.15182259887483695461833666199, 7.76771279224535161954925552598