| L(s) = 1 | − 0.903·2-s + 3.04·3-s − 1.18·4-s − 5-s − 2.75·6-s + 7-s + 2.87·8-s + 6.27·9-s + 0.903·10-s − 4.64·11-s − 3.60·12-s + 1.52·13-s − 0.903·14-s − 3.04·15-s − 0.232·16-s + 0.814·17-s − 5.67·18-s − 3.90·19-s + 1.18·20-s + 3.04·21-s + 4.19·22-s − 0.527·23-s + 8.76·24-s + 25-s − 1.37·26-s + 9.97·27-s − 1.18·28-s + ⋯ |
| L(s) = 1 | − 0.639·2-s + 1.75·3-s − 0.591·4-s − 0.447·5-s − 1.12·6-s + 0.377·7-s + 1.01·8-s + 2.09·9-s + 0.285·10-s − 1.40·11-s − 1.04·12-s + 0.422·13-s − 0.241·14-s − 0.786·15-s − 0.0582·16-s + 0.197·17-s − 1.33·18-s − 0.895·19-s + 0.264·20-s + 0.664·21-s + 0.894·22-s − 0.110·23-s + 1.78·24-s + 0.200·25-s − 0.269·26-s + 1.91·27-s − 0.223·28-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 | \( 1 + 0.903T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 - 0.814T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 23 | \( 1 + 0.527T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.96T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.84078054410322793990965555404, −7.33743578415581570573240075006, −6.33053706575055669923777508555, −4.97702448974655483938809233701, −4.65758180913583373337401643700, −3.67981827528524476558151390670, −3.14362380835843816440578707613, −2.17891750843776077645709707800, −1.41481639966664848433003199499, 0,
1.41481639966664848433003199499, 2.17891750843776077645709707800, 3.14362380835843816440578707613, 3.67981827528524476558151390670, 4.65758180913583373337401643700, 4.97702448974655483938809233701, 6.33053706575055669923777508555, 7.33743578415581570573240075006, 7.84078054410322793990965555404
