| L(s) = 1 | − 1.22·2-s − 2.22·3-s − 0.498·4-s + 5-s + 2.72·6-s − 7-s + 3.06·8-s + 1.94·9-s − 1.22·10-s − 3.57·11-s + 1.10·12-s + 4.39·13-s + 1.22·14-s − 2.22·15-s − 2.75·16-s − 3.48·17-s − 2.37·18-s + 6.31·19-s − 0.498·20-s + 2.22·21-s + 4.38·22-s + 1.03·23-s − 6.80·24-s + 25-s − 5.38·26-s + 2.35·27-s + 0.498·28-s + ⋯ |
| L(s) = 1 | − 0.866·2-s − 1.28·3-s − 0.249·4-s + 0.447·5-s + 1.11·6-s − 0.377·7-s + 1.08·8-s + 0.647·9-s − 0.387·10-s − 1.07·11-s + 0.319·12-s + 1.21·13-s + 0.327·14-s − 0.573·15-s − 0.688·16-s − 0.845·17-s − 0.560·18-s + 1.44·19-s − 0.111·20-s + 0.485·21-s + 0.934·22-s + 0.215·23-s − 1.38·24-s + 0.200·25-s − 1.05·26-s + 0.452·27-s + 0.0941·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 3 | \( 1 + 2.22T + 3T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 - 1.03T + 23T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 - 0.525T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 0.272T + 67T^{2} \) |
| 71 | \( 1 + 3.99T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 6.01T + 97T^{2} \) |
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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
−7.48483996553386415414238807905, −6.81751250854119134224971379303, −6.07422861813375034012108160577, −5.37330473402887510837065449206, −5.02298762373239825813777683678, −4.01338973655119625982438468422, −3.05992757062252584521696809020, −1.81989058982575962472750252555, −0.895725872865292224533911256165, 0,
0.895725872865292224533911256165, 1.81989058982575962472750252555, 3.05992757062252584521696809020, 4.01338973655119625982438468422, 5.02298762373239825813777683678, 5.37330473402887510837065449206, 6.07422861813375034012108160577, 6.81751250854119134224971379303, 7.48483996553386415414238807905
