L(s) = 1 | − 0.978·2-s − 3.27·3-s − 1.04·4-s − 5-s + 3.20·6-s − 7-s + 2.97·8-s + 7.74·9-s + 0.978·10-s + 6.25·11-s + 3.41·12-s + 0.406·13-s + 0.978·14-s + 3.27·15-s − 0.824·16-s + 6.95·17-s − 7.57·18-s + 0.159·19-s + 1.04·20-s + 3.27·21-s − 6.11·22-s + 3.08·23-s − 9.75·24-s + 25-s − 0.397·26-s − 15.5·27-s + 1.04·28-s + ⋯ |
L(s) = 1 | − 0.691·2-s − 1.89·3-s − 0.521·4-s − 0.447·5-s + 1.30·6-s − 0.377·7-s + 1.05·8-s + 2.58·9-s + 0.309·10-s + 1.88·11-s + 0.987·12-s + 0.112·13-s + 0.261·14-s + 0.846·15-s − 0.206·16-s + 1.68·17-s − 1.78·18-s + 0.0365·19-s + 0.233·20-s + 0.715·21-s − 1.30·22-s + 0.644·23-s − 1.99·24-s + 0.200·25-s − 0.0779·26-s − 2.99·27-s + 0.197·28-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 | \( 1 + 0.978T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 11 | \( 1 - 6.25T + 11T^{2} \) |
| 13 | \( 1 - 0.406T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 - 0.159T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 0.474T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 + 0.713T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 7.43T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 - 9.37T + 89T^{2} \) |
| 97 | \( 1 + 8.66T + 97T^{2} \) |
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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
−7.34501286801389843320440235855, −6.77533589077562918196920821887, −6.19178530059376491735551176010, −5.39858268526693721335745224668, −4.79967031451653126972324228973, −4.02586228816774473143885032306, −3.47619487176962939160763106049, −1.38071603572512385727191989718, −1.08529431653670817187799754898, 0,
1.08529431653670817187799754898, 1.38071603572512385727191989718, 3.47619487176962939160763106049, 4.02586228816774473143885032306, 4.79967031451653126972324228973, 5.39858268526693721335745224668, 6.19178530059376491735551176010, 6.77533589077562918196920821887, 7.34501286801389843320440235855