| L(s) = 1 | − 1.46·2-s − 3-s + 0.137·4-s − 2.88·5-s + 1.46·6-s + 1.16·7-s + 2.72·8-s + 9-s + 4.21·10-s − 4.61·11-s − 0.137·12-s − 13-s − 1.69·14-s + 2.88·15-s − 4.25·16-s + 4.00·17-s − 1.46·18-s + 3.52·19-s − 0.396·20-s − 1.16·21-s + 6.74·22-s − 0.748·23-s − 2.72·24-s + 3.32·25-s + 1.46·26-s − 27-s + 0.159·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0686·4-s − 1.29·5-s + 0.596·6-s + 0.439·7-s + 0.962·8-s + 0.333·9-s + 1.33·10-s − 1.39·11-s − 0.0396·12-s − 0.277·13-s − 0.453·14-s + 0.745·15-s − 1.06·16-s + 0.971·17-s − 0.344·18-s + 0.808·19-s − 0.0885·20-s − 0.253·21-s + 1.43·22-s − 0.156·23-s − 0.555·24-s + 0.665·25-s + 0.286·26-s − 0.192·27-s + 0.0301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + 0.748T + 23T^{2} \) |
| 29 | \( 1 + 9.69T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 5.13T + 53T^{2} \) |
| 59 | \( 1 - 9.67T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.91T + 71T^{2} \) |
| 73 | \( 1 + 0.273T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 3.13T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.003092528221746844774611647056, −7.38263275971099017776845923747, −7.30720905414680933040018513636, −5.60859739823144869498630331979, −5.22778406092859841649386232828, −4.25869319169484387989075167577, −3.54270220698423101889896988514, −2.20283503329499772629997402708, −0.909342632479748684047450347095, 0,
0.909342632479748684047450347095, 2.20283503329499772629997402708, 3.54270220698423101889896988514, 4.25869319169484387989075167577, 5.22778406092859841649386232828, 5.60859739823144869498630331979, 7.30720905414680933040018513636, 7.38263275971099017776845923747, 8.003092528221746844774611647056
