| L(s) = 1 | + 0.0379·2-s − 3.37·3-s − 1.99·4-s + 5-s − 0.128·6-s + 2.89·7-s − 0.151·8-s + 8.40·9-s + 0.0379·10-s + 11-s + 6.74·12-s + 0.109·14-s − 3.37·15-s + 3.99·16-s − 0.920·17-s + 0.318·18-s + 4.05·19-s − 1.99·20-s − 9.76·21-s + 0.0379·22-s + 2.61·23-s + 0.512·24-s + 25-s − 18.2·27-s − 5.77·28-s + 9.54·29-s − 0.128·30-s + ⋯ |
| L(s) = 1 | + 0.0268·2-s − 1.94·3-s − 0.999·4-s + 0.447·5-s − 0.0523·6-s + 1.09·7-s − 0.0536·8-s + 2.80·9-s + 0.0120·10-s + 0.301·11-s + 1.94·12-s + 0.0293·14-s − 0.871·15-s + 0.997·16-s − 0.223·17-s + 0.0751·18-s + 0.929·19-s − 0.446·20-s − 2.12·21-s + 0.00809·22-s + 0.545·23-s + 0.104·24-s + 0.200·25-s − 3.51·27-s − 1.09·28-s + 1.77·29-s − 0.0234·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.0379T + 2T^{2} \) |
| 3 | \( 1 + 3.37T + 3T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 17 | \( 1 + 0.920T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 - 9.54T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 0.451T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 + 4.96T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.30092046521317422606328981066, −6.44997012213165949093146505496, −5.89100786650838314482173684849, −5.14862740573744388745995895041, −4.82183609201465490901080935150, −4.34224373552766059684236846240, −3.24718820752940527497315042040, −1.58391647773598017841540939553, −1.14748032065072459559819035589, 0,
1.14748032065072459559819035589, 1.58391647773598017841540939553, 3.24718820752940527497315042040, 4.34224373552766059684236846240, 4.82183609201465490901080935150, 5.14862740573744388745995895041, 5.89100786650838314482173684849, 6.44997012213165949093146505496, 7.30092046521317422606328981066
