| L(s) = 1 | + 2-s + 2.50·3-s + 4-s + 2.19·5-s + 2.50·6-s − 4.76·7-s + 8-s + 3.27·9-s + 2.19·10-s − 0.844·11-s + 2.50·12-s + 0.968·13-s − 4.76·14-s + 5.49·15-s + 16-s − 0.386·17-s + 3.27·18-s − 19-s + 2.19·20-s − 11.9·21-s − 0.844·22-s + 7.90·23-s + 2.50·24-s − 0.184·25-s + 0.968·26-s + 0.681·27-s − 4.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.44·3-s + 0.5·4-s + 0.981·5-s + 1.02·6-s − 1.79·7-s + 0.353·8-s + 1.09·9-s + 0.693·10-s − 0.254·11-s + 0.722·12-s + 0.268·13-s − 1.27·14-s + 1.41·15-s + 0.250·16-s − 0.0937·17-s + 0.771·18-s − 0.229·19-s + 0.490·20-s − 2.60·21-s − 0.180·22-s + 1.64·23-s + 0.511·24-s − 0.0369·25-s + 0.190·26-s + 0.131·27-s − 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.894460644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.894460644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 0.844T + 11T^{2} \) |
| 13 | \( 1 - 0.968T + 13T^{2} \) |
| 17 | \( 1 + 0.386T + 17T^{2} \) |
| 23 | \( 1 - 7.90T + 23T^{2} \) |
| 29 | \( 1 - 0.136T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 0.267T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 + 9.60T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.82542833640884969997149851634, −6.97178277005873766415944580329, −6.42643928310216023774031250384, −5.91206072115321861467463919151, −4.98727503980578921472405292884, −4.01041534429845525117944652121, −3.37045711878266166647168552792, −2.62415651050765066674682863700, −2.42385462548868354092783739448, −1.01779383047814929331319519462,
1.01779383047814929331319519462, 2.42385462548868354092783739448, 2.62415651050765066674682863700, 3.37045711878266166647168552792, 4.01041534429845525117944652121, 4.98727503980578921472405292884, 5.91206072115321861467463919151, 6.42643928310216023774031250384, 6.97178277005873766415944580329, 7.82542833640884969997149851634
