| L(s) = 1 | + 0.823·2-s − 1.32·4-s − 2.98·5-s + 7-s − 2.73·8-s − 2.45·10-s + 2.20·13-s + 0.823·14-s + 0.390·16-s − 4.40·17-s − 1.72·19-s + 3.94·20-s + 8.39·23-s + 3.91·25-s + 1.81·26-s − 1.32·28-s − 3.29·29-s + 7.47·31-s + 5.79·32-s − 3.62·34-s − 2.98·35-s − 8.78·37-s − 1.42·38-s + 8.16·40-s − 5.39·41-s + 9.44·43-s + 6.91·46-s + ⋯ |
| L(s) = 1 | + 0.582·2-s − 0.660·4-s − 1.33·5-s + 0.377·7-s − 0.967·8-s − 0.777·10-s + 0.610·13-s + 0.220·14-s + 0.0976·16-s − 1.06·17-s − 0.395·19-s + 0.882·20-s + 1.75·23-s + 0.782·25-s + 0.355·26-s − 0.249·28-s − 0.611·29-s + 1.34·31-s + 1.02·32-s − 0.622·34-s − 0.504·35-s − 1.44·37-s − 0.230·38-s + 1.29·40-s − 0.842·41-s + 1.44·43-s + 1.01·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.823T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 4.40T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 + 7.63T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.50636476970555516208949140077, −6.85001121977700711012980445196, −6.09596813458985085858751713694, −5.10804824512119760648076539889, −4.66862225508484290447135747638, −3.95277471070987153244283923722, −3.45211543402027363857502811474, −2.51598232586754833141601771693, −1.06249706614282958018771635714, 0,
1.06249706614282958018771635714, 2.51598232586754833141601771693, 3.45211543402027363857502811474, 3.95277471070987153244283923722, 4.66862225508484290447135747638, 5.10804824512119760648076539889, 6.09596813458985085858751713694, 6.85001121977700711012980445196, 7.50636476970555516208949140077
