| L(s) = 1 | − 1.59·2-s − 1.35·3-s + 0.535·4-s + 2.16·6-s + 0.175·7-s + 2.33·8-s − 1.15·9-s + 11-s − 0.726·12-s + 1.01·13-s − 0.279·14-s − 4.78·16-s + 3.69·17-s + 1.84·18-s + 19-s − 0.238·21-s − 1.59·22-s − 6.06·23-s − 3.16·24-s − 1.61·26-s + 5.64·27-s + 0.0940·28-s + 3.54·29-s − 0.962·31-s + 2.95·32-s − 1.35·33-s − 5.87·34-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.783·3-s + 0.267·4-s + 0.882·6-s + 0.0664·7-s + 0.824·8-s − 0.385·9-s + 0.301·11-s − 0.209·12-s + 0.281·13-s − 0.0747·14-s − 1.19·16-s + 0.895·17-s + 0.434·18-s + 0.229·19-s − 0.0520·21-s − 0.339·22-s − 1.26·23-s − 0.646·24-s − 0.316·26-s + 1.08·27-s + 0.0177·28-s + 0.658·29-s − 0.172·31-s + 0.521·32-s − 0.236·33-s − 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6178180452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6178180452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 0.175T + 7T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 23 | \( 1 + 6.06T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 + 0.962T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 - 0.398T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 - 9.15T + 67T^{2} \) |
| 71 | \( 1 - 7.33T + 71T^{2} \) |
| 73 | \( 1 + 1.20T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 0.412T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−8.157699133572537271676511608847, −7.78444155308005319853883752545, −6.81332218211942682085354482438, −6.12741238454561269320660603219, −5.43916945575632247005696828357, −4.61390368334187826979546615273, −3.77678168603762581422843017890, −2.60960329894591175044921170035, −1.42916102230389665868282134645, −0.57409029324842722073526159438,
0.57409029324842722073526159438, 1.42916102230389665868282134645, 2.60960329894591175044921170035, 3.77678168603762581422843017890, 4.61390368334187826979546615273, 5.43916945575632247005696828357, 6.12741238454561269320660603219, 6.81332218211942682085354482438, 7.78444155308005319853883752545, 8.157699133572537271676511608847
