| L(s) = 1 | + 2-s + 4-s + 5-s + 1.15·7-s + 8-s + 10-s + 6.34·11-s − 5.23·13-s + 1.15·14-s + 16-s + 6.50·17-s + 1.05·19-s + 20-s + 6.34·22-s − 0.797·23-s + 25-s − 5.23·26-s + 1.15·28-s + 7.76·29-s + 9.54·31-s + 32-s + 6.50·34-s + 1.15·35-s + 0.264·37-s + 1.05·38-s + 40-s − 0.200·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.437·7-s + 0.353·8-s + 0.316·10-s + 1.91·11-s − 1.45·13-s + 0.309·14-s + 0.250·16-s + 1.57·17-s + 0.241·19-s + 0.223·20-s + 1.35·22-s − 0.166·23-s + 0.200·25-s − 1.02·26-s + 0.218·28-s + 1.44·29-s + 1.71·31-s + 0.176·32-s + 1.11·34-s + 0.195·35-s + 0.0434·37-s + 0.170·38-s + 0.158·40-s − 0.0313·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.682910874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.682910874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 7 | \( 1 - 1.15T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 + 0.797T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 9.54T + 31T^{2} \) |
| 37 | \( 1 - 0.264T + 37T^{2} \) |
| 41 | \( 1 + 0.200T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 8.25T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 - 0.830T + 83T^{2} \) |
| 97 | \( 1 - 0.404T + 97T^{2} \) |
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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
−7.80018400709230496381440920067, −6.91359523982411470554571486547, −6.42713017043194945879715969054, −5.78025666564543854916363719682, −4.71400210836087639739770847495, −4.65396220931192158372612518664, −3.43860406587292545635236287137, −2.86752211804384052191616813361, −1.75301294552548741362429462637, −1.06090653682844516953433955298,
1.06090653682844516953433955298, 1.75301294552548741362429462637, 2.86752211804384052191616813361, 3.43860406587292545635236287137, 4.65396220931192158372612518664, 4.71400210836087639739770847495, 5.78025666564543854916363719682, 6.42713017043194945879715969054, 6.91359523982411470554571486547, 7.80018400709230496381440920067
