| L(s) = 1 | − 0.0913·2-s − 3-s − 1.99·4-s − 1.96·5-s + 0.0913·6-s − 3.33·7-s + 0.364·8-s + 9-s + 0.179·10-s − 2.42·11-s + 1.99·12-s + 3.46·13-s + 0.304·14-s + 1.96·15-s + 3.94·16-s − 17-s − 0.0913·18-s − 3.16·19-s + 3.91·20-s + 3.33·21-s + 0.221·22-s + 1.46·23-s − 0.364·24-s − 1.12·25-s − 0.316·26-s − 27-s + 6.63·28-s + ⋯ |
| L(s) = 1 | − 0.0646·2-s − 0.577·3-s − 0.995·4-s − 0.879·5-s + 0.0373·6-s − 1.25·7-s + 0.128·8-s + 0.333·9-s + 0.0568·10-s − 0.731·11-s + 0.574·12-s + 0.961·13-s + 0.0813·14-s + 0.508·15-s + 0.987·16-s − 0.242·17-s − 0.0215·18-s − 0.727·19-s + 0.876·20-s + 0.726·21-s + 0.0472·22-s + 0.305·23-s − 0.0744·24-s − 0.225·25-s − 0.0621·26-s − 0.192·27-s + 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0913T + 2T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 7.83T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 + 3.41T + 73T^{2} \) |
| 83 | \( 1 + 0.840T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−8.038743010237740293660621974286, −7.50117778381677626036801104117, −6.39421005274721452227307118194, −5.98835929267459769318267228606, −4.99188886304572072280627332521, −4.17719138727562705698649429606, −3.67477676561711573461880863994, −2.67272461132259342038252021654, −0.890279982693816704402053515583, 0,
0.890279982693816704402053515583, 2.67272461132259342038252021654, 3.67477676561711573461880863994, 4.17719138727562705698649429606, 4.99188886304572072280627332521, 5.98835929267459769318267228606, 6.39421005274721452227307118194, 7.50117778381677626036801104117, 8.038743010237740293660621974286
