| L(s) = 1 | − 3.31i·5-s − 4.17·7-s − 5.67i·11-s + i·13-s + 1.15·17-s − 7.00i·19-s − 5.89·23-s − 5.98·25-s + 1.07i·29-s − 3.76·31-s + 13.8i·35-s − 1.08i·37-s + 4.24·41-s − 4.23i·43-s + 1.13·47-s + ⋯ |
| L(s) = 1 | − 1.48i·5-s − 1.57·7-s − 1.71i·11-s + 0.277i·13-s + 0.278·17-s − 1.60i·19-s − 1.22·23-s − 1.19·25-s + 0.199i·29-s − 0.677·31-s + 2.34i·35-s − 0.178i·37-s + 0.662·41-s − 0.646i·43-s + 0.165·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5503715174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5503715174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 + 5.67iT - 11T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 7.00iT - 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 - 1.07iT - 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 + 1.08iT - 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 4.23iT - 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 + 8.20iT - 53T^{2} \) |
| 59 | \( 1 - 8.89iT - 59T^{2} \) |
| 61 | \( 1 - 1.74iT - 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 7.03iT - 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 5.44T + 97T^{2} \) |
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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
−8.283231875968830708664670235782, −7.29056227532792876676615670772, −6.40085547521867468503882138934, −5.79592911514386060488919637088, −5.13919536448391237399254614383, −4.07579689511991014302640473393, −3.43118206985040343083379487159, −2.44273635747056585371158768677, −0.923697689349561300239674471997, −0.19199743655628374169696922445,
1.87837356456569113079547302474, 2.70889753505689209688327947524, 3.54804392462398926887780535011, 4.09705118464173252161301285365, 5.44219412841876130628106505968, 6.39811768775256637811813365349, 6.52891873340030949388777603727, 7.58292004396600723913361700498, 7.83683338290748890572875043559, 9.372433604642288720848810067549
