L(s) = 1 | + (−1.70 + 0.292i)3-s + (−2 − 2i)7-s + (2.82 − i)9-s − 5.65i·11-s + (−2.82 + 2.82i)17-s + 4i·19-s + (4 + 2.82i)21-s + (4.24 + 4.24i)23-s + (−4.53 + 2.53i)27-s + 5.65·29-s − 8·31-s + (1.65 + 9.65i)33-s + (−8 − 8i)37-s + 5.65i·41-s + (−2 + 2i)43-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.169i)3-s + (−0.755 − 0.755i)7-s + (0.942 − 0.333i)9-s − 1.70i·11-s + (−0.685 + 0.685i)17-s + 0.917i·19-s + (0.872 + 0.617i)21-s + (0.884 + 0.884i)23-s + (−0.872 + 0.487i)27-s + 1.05·29-s − 1.43·31-s + (0.288 + 1.68i)33-s + (−1.31 − 1.31i)37-s + 0.883i·41-s + (−0.304 + 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (8 + 8i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2 - 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.65 + 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (6 + 6i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (8 - 8i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (-9.89 - 9.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-8 - 8i)T + 97iT^{2} \) |
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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
−9.408721572497026155637185590260, −8.548203821529180701599916673453, −7.47806463586219298492644490378, −6.59456570505574244109479947746, −5.98950487083743241321939682913, −5.16720725357775466167810705883, −3.91058196048793774274400447395, −3.32330537809552337108992570988, −1.33915074520200397122627638469, 0,
1.83313232375084836687615780898, 2.95623639703052422194080173491, 4.60051738945155253587241994299, 4.95349925205421325153156435037, 6.14338251429471009493355995330, 6.90312460301925017665630973448, 7.33141066806559101920320673994, 8.828958013841248821221912085448, 9.402822132059400201473778505126