| L(s) = 1 | − 2-s − 2.65·3-s + 4-s + 0.987·5-s + 2.65·6-s + 3.23·7-s − 8-s + 4.03·9-s − 0.987·10-s − 4.10·11-s − 2.65·12-s − 3.23·14-s − 2.61·15-s + 16-s + 4.83·17-s − 4.03·18-s + 19-s + 0.987·20-s − 8.59·21-s + 4.10·22-s + 3.74·23-s + 2.65·24-s − 4.02·25-s − 2.75·27-s + 3.23·28-s − 7.96·29-s + 2.61·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.441·5-s + 1.08·6-s + 1.22·7-s − 0.353·8-s + 1.34·9-s − 0.312·10-s − 1.23·11-s − 0.765·12-s − 0.865·14-s − 0.676·15-s + 0.250·16-s + 1.17·17-s − 0.951·18-s + 0.229·19-s + 0.220·20-s − 1.87·21-s + 0.874·22-s + 0.781·23-s + 0.541·24-s − 0.805·25-s − 0.529·27-s + 0.612·28-s − 1.47·29-s + 0.478·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.65T + 3T^{2} \) |
| 5 | \( 1 - 0.987T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 0.529T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 - 4.04T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 0.186T + 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−7.54096574845235821399576517154, −7.20946302358258910532623780471, −5.98272332416948278348304393275, −5.61675007358526896824846423505, −5.14712420632298805846267876070, −4.30761831215392853105704556612, −3.00476616928077212839968529686, −1.88608530095745235234511186117, −1.14529209964374965378553416421, 0,
1.14529209964374965378553416421, 1.88608530095745235234511186117, 3.00476616928077212839968529686, 4.30761831215392853105704556612, 5.14712420632298805846267876070, 5.61675007358526896824846423505, 5.98272332416948278348304393275, 7.20946302358258910532623780471, 7.54096574845235821399576517154
