| L(s) = 1 | − 0.244·2-s − 2.73·3-s − 1.94·4-s + 0.669·6-s + 1.94·7-s + 0.963·8-s + 4.49·9-s + 4.23·11-s + 5.31·12-s − 1.26·13-s − 0.474·14-s + 3.64·16-s + 2.46·17-s − 1.09·18-s − 5.31·21-s − 1.03·22-s + 5.13·23-s − 2.63·24-s + 0.310·26-s − 4.10·27-s − 3.76·28-s − 8.64·29-s − 5.10·31-s − 2.81·32-s − 11.5·33-s − 0.601·34-s − 8.72·36-s + ⋯ |
| L(s) = 1 | − 0.172·2-s − 1.58·3-s − 0.970·4-s + 0.273·6-s + 0.733·7-s + 0.340·8-s + 1.49·9-s + 1.27·11-s + 1.53·12-s − 0.352·13-s − 0.126·14-s + 0.911·16-s + 0.596·17-s − 0.259·18-s − 1.15·21-s − 0.220·22-s + 1.07·23-s − 0.538·24-s + 0.0608·26-s − 0.789·27-s − 0.711·28-s − 1.60·29-s − 0.916·31-s − 0.498·32-s − 2.01·33-s − 0.103·34-s − 1.45·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.244T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 - 0.689T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 - 2.80T + 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 + 0.791T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 6.06T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−7.34406111344880824748960373316, −6.63796237545735853827629223343, −5.84010933158812661054490996580, −5.33075046280362755237286076985, −4.72133228302019739709150170787, −4.16506733713406381908988164681, −3.32068784009607381426167538050, −1.65599738160685204754258464746, −1.05767867253684971217756417981, 0,
1.05767867253684971217756417981, 1.65599738160685204754258464746, 3.32068784009607381426167538050, 4.16506733713406381908988164681, 4.72133228302019739709150170787, 5.33075046280362755237286076985, 5.84010933158812661054490996580, 6.63796237545735853827629223343, 7.34406111344880824748960373316
