| L(s) = 1 | − 2-s − 3.31·3-s + 4-s − 0.532·5-s + 3.31·6-s − 2.51·7-s − 8-s + 7.99·9-s + 0.532·10-s + 6.00·11-s − 3.31·12-s + 4.31·13-s + 2.51·14-s + 1.76·15-s + 16-s + 2.00·17-s − 7.99·18-s − 19-s − 0.532·20-s + 8.32·21-s − 6.00·22-s − 2.59·23-s + 3.31·24-s − 4.71·25-s − 4.31·26-s − 16.5·27-s − 2.51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.91·3-s + 0.5·4-s − 0.238·5-s + 1.35·6-s − 0.949·7-s − 0.353·8-s + 2.66·9-s + 0.168·10-s + 1.80·11-s − 0.957·12-s + 1.19·13-s + 0.671·14-s + 0.455·15-s + 0.250·16-s + 0.485·17-s − 1.88·18-s − 0.229·19-s − 0.119·20-s + 1.81·21-s − 1.27·22-s − 0.540·23-s + 0.676·24-s − 0.943·25-s − 0.846·26-s − 3.18·27-s − 0.474·28-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 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 + 0.532T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 - 6.00T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 0.879T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 7.84T + 97T^{2} \) |
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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
−7.20336359622919751035887992058, −6.49818491143245938561302279651, −6.32845277542448071290999033297, −5.81130724699252900984609955776, −4.68385730577017968817921389708, −3.99727568492586585691199179429, −3.29045717730358073923166023330, −1.59023857009934958213518791542, −1.03631530460901430805135592834, 0,
1.03631530460901430805135592834, 1.59023857009934958213518791542, 3.29045717730358073923166023330, 3.99727568492586585691199179429, 4.68385730577017968817921389708, 5.81130724699252900984609955776, 6.32845277542448071290999033297, 6.49818491143245938561302279651, 7.20336359622919751035887992058
