| L(s) = 1 | + 2-s + 2.76·3-s + 4-s − 1.78·5-s + 2.76·6-s − 4.04·7-s + 8-s + 4.64·9-s − 1.78·10-s − 1.77·11-s + 2.76·12-s + 2.72·13-s − 4.04·14-s − 4.93·15-s + 16-s − 2.11·17-s + 4.64·18-s + 19-s − 1.78·20-s − 11.1·21-s − 1.77·22-s + 3.01·23-s + 2.76·24-s − 1.81·25-s + 2.72·26-s + 4.56·27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.59·3-s + 0.5·4-s − 0.797·5-s + 1.12·6-s − 1.52·7-s + 0.353·8-s + 1.54·9-s − 0.564·10-s − 0.534·11-s + 0.798·12-s + 0.757·13-s − 1.08·14-s − 1.27·15-s + 0.250·16-s − 0.512·17-s + 1.09·18-s + 0.229·19-s − 0.398·20-s − 2.44·21-s − 0.378·22-s + 0.628·23-s + 0.564·24-s − 0.363·25-s + 0.535·26-s + 0.877·27-s − 0.764·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.0551T + 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 4.45T + 61T^{2} \) |
| 67 | \( 1 - 0.780T + 67T^{2} \) |
| 71 | \( 1 - 0.868T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 + 8.77T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 5.44T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.45099823226925950744074265691, −6.97869875966928667174477566775, −6.17244258923802144958602506143, −5.36492906280660633736164127065, −4.21150265664622431610191891131, −3.74849517274626114579644015414, −3.19194598856328121486750339874, −2.67290641435814870264133151292, −1.65104188744116694953331636561, 0,
1.65104188744116694953331636561, 2.67290641435814870264133151292, 3.19194598856328121486750339874, 3.74849517274626114579644015414, 4.21150265664622431610191891131, 5.36492906280660633736164127065, 6.17244258923802144958602506143, 6.97869875966928667174477566775, 7.45099823226925950744074265691
