L(s) = 1 | + 2.65·2-s + 3-s + 5.03·4-s − 1.41·5-s + 2.65·6-s − 1.11·7-s + 8.06·8-s + 9-s − 3.74·10-s − 4.97·11-s + 5.03·12-s − 6.21·13-s − 2.94·14-s − 1.41·15-s + 11.3·16-s + 17-s + 2.65·18-s − 6.01·19-s − 7.10·20-s − 1.11·21-s − 13.2·22-s − 2.55·23-s + 8.06·24-s − 3.00·25-s − 16.4·26-s + 27-s − 5.59·28-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.51·4-s − 0.630·5-s + 1.08·6-s − 0.420·7-s + 2.85·8-s + 0.333·9-s − 1.18·10-s − 1.50·11-s + 1.45·12-s − 1.72·13-s − 0.787·14-s − 0.364·15-s + 2.82·16-s + 0.242·17-s + 0.625·18-s − 1.37·19-s − 1.58·20-s − 0.242·21-s − 2.81·22-s − 0.531·23-s + 1.64·24-s − 0.601·25-s − 3.23·26-s + 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 + 6.21T + 13T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 9.40T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.36111373321263966933076361246, −6.67494499141198108513090657304, −6.01133855146270403158192456263, −5.03034063143130446863867091281, −4.72035015927478672139708682890, −3.97695423193561093116529009101, −3.13832790006707892289190306372, −2.59605267218609821845513946926, −2.00875042228768899834190672468, 0,
2.00875042228768899834190672468, 2.59605267218609821845513946926, 3.13832790006707892289190306372, 3.97695423193561093116529009101, 4.72035015927478672139708682890, 5.03034063143130446863867091281, 6.01133855146270403158192456263, 6.67494499141198108513090657304, 7.36111373321263966933076361246