| L(s) = 1 | + 0.347·2-s − 1.87·4-s + 1.34·5-s − 0.652·7-s − 1.34·8-s + 0.467·10-s − 3.94·11-s + 1.69·13-s − 0.226·14-s + 3.29·16-s + 3.83·17-s − 2.53·20-s − 1.36·22-s + 3.63·23-s − 3.18·25-s + 0.588·26-s + 1.22·28-s − 10.5·29-s + 6.46·31-s + 3.83·32-s + 1.33·34-s − 0.879·35-s − 2.94·37-s − 1.81·40-s + 1.50·41-s − 9.36·43-s + 7.41·44-s + ⋯ |
| L(s) = 1 | + 0.245·2-s − 0.939·4-s + 0.602·5-s − 0.246·7-s − 0.476·8-s + 0.147·10-s − 1.18·11-s + 0.469·13-s − 0.0605·14-s + 0.822·16-s + 0.930·17-s − 0.566·20-s − 0.291·22-s + 0.758·23-s − 0.636·25-s + 0.115·26-s + 0.231·28-s − 1.95·29-s + 1.16·31-s + 0.678·32-s + 0.228·34-s − 0.148·35-s − 0.483·37-s − 0.287·40-s + 0.235·41-s − 1.42·43-s + 1.11·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529918007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529918007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 - 4.31T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 4.50T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−8.661528088526571561060020028086, −8.001754309133772974814996416618, −7.23789188937206170387185495946, −6.12585299449138411813521988457, −5.47991933697402943491407553552, −5.03598891302171777081717378595, −3.89008569540406338779097759464, −3.21557383356775530205009893249, −2.11402975531072306752974516724, −0.71400243975855102173931326550,
0.71400243975855102173931326550, 2.11402975531072306752974516724, 3.21557383356775530205009893249, 3.89008569540406338779097759464, 5.03598891302171777081717378595, 5.47991933697402943491407553552, 6.12585299449138411813521988457, 7.23789188937206170387185495946, 8.001754309133772974814996416618, 8.661528088526571561060020028086
