| L(s) = 1 | − 1.69·2-s + 3.34·3-s + 0.867·4-s − 5-s − 5.65·6-s − 3.70·7-s + 1.91·8-s + 8.16·9-s + 1.69·10-s + 11-s + 2.89·12-s + 2.41·13-s + 6.27·14-s − 3.34·15-s − 4.98·16-s − 3.23·17-s − 13.8·18-s − 5.91·19-s − 0.867·20-s − 12.3·21-s − 1.69·22-s + 7.82·23-s + 6.40·24-s + 25-s − 4.08·26-s + 17.2·27-s − 3.21·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 1.92·3-s + 0.433·4-s − 0.447·5-s − 2.31·6-s − 1.40·7-s + 0.677·8-s + 2.72·9-s + 0.535·10-s + 0.301·11-s + 0.837·12-s + 0.669·13-s + 1.67·14-s − 0.862·15-s − 1.24·16-s − 0.785·17-s − 3.26·18-s − 1.35·19-s − 0.194·20-s − 2.70·21-s − 0.361·22-s + 1.63·23-s + 1.30·24-s + 0.200·25-s − 0.801·26-s + 3.32·27-s − 0.607·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 - 3.34T + 3T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 + 8.31T + 29T^{2} \) |
| 31 | \( 1 + 9.83T + 31T^{2} \) |
| 37 | \( 1 + 7.10T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 6.64T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 3.18T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 + 0.203T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.498203687306106421062787108140, −7.38332420771802769540096755045, −7.16570175661092037769410861889, −6.37865055413987911404843181470, −4.75644125502193987795022787353, −3.73929101073381572760636105313, −3.47683478077218597111014012216, −2.32758896854015225731451741855, −1.51894052435207602438595197105, 0,
1.51894052435207602438595197105, 2.32758896854015225731451741855, 3.47683478077218597111014012216, 3.73929101073381572760636105313, 4.75644125502193987795022787353, 6.37865055413987911404843181470, 7.16570175661092037769410861889, 7.38332420771802769540096755045, 8.498203687306106421062787108140
