| L(s) = 1 | − 2.59·2-s + 0.990·3-s + 4.73·4-s + 5-s − 2.57·6-s + 7-s − 7.09·8-s − 2.01·9-s − 2.59·10-s + 2.69·11-s + 4.68·12-s − 1.27·13-s − 2.59·14-s + 0.990·15-s + 8.94·16-s + 3.50·17-s + 5.23·18-s − 2.44·19-s + 4.73·20-s + 0.990·21-s − 6.98·22-s + 2.80·23-s − 7.02·24-s + 25-s + 3.31·26-s − 4.97·27-s + 4.73·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.571·3-s + 2.36·4-s + 0.447·5-s − 1.04·6-s + 0.377·7-s − 2.50·8-s − 0.673·9-s − 0.820·10-s + 0.812·11-s + 1.35·12-s − 0.354·13-s − 0.693·14-s + 0.255·15-s + 2.23·16-s + 0.849·17-s + 1.23·18-s − 0.561·19-s + 1.05·20-s + 0.216·21-s − 1.49·22-s + 0.584·23-s − 1.43·24-s + 0.200·25-s + 0.651·26-s − 0.956·27-s + 0.894·28-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 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 0.990T + 3T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 2.80T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 2.10T + 53T^{2} \) |
| 59 | \( 1 + 0.734T + 59T^{2} \) |
| 61 | \( 1 + 0.871T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 0.0533T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−7.56651734465572699945944459685, −7.23301739014683628362169188623, −6.29733403040073222107999433952, −5.78387603534962861767283546233, −4.72870473862082841787625825669, −3.44974056589357202665829458801, −2.78873143502099117628659909022, −1.89900898778894974738987050691, −1.29339847581856791397141887418, 0,
1.29339847581856791397141887418, 1.89900898778894974738987050691, 2.78873143502099117628659909022, 3.44974056589357202665829458801, 4.72870473862082841787625825669, 5.78387603534962861767283546233, 6.29733403040073222107999433952, 7.23301739014683628362169188623, 7.56651734465572699945944459685
