| L(s) = 1 | + (1.61 + 0.618i)3-s + (0.381 − 2.61i)7-s + (2.23 + 2.00i)9-s − 5.23i·11-s + 3.23·13-s − 4.47i·17-s + 2.76i·19-s + (2.23 − 4i)21-s − 5.70·23-s + (2.38 + 4.61i)27-s − 4i·29-s − 1.23i·31-s + (3.23 − 8.47i)33-s − 4.47i·37-s + (5.23 + 2.00i)39-s + ⋯ |
| L(s) = 1 | + (0.934 + 0.356i)3-s + (0.144 − 0.989i)7-s + (0.745 + 0.666i)9-s − 1.57i·11-s + 0.897·13-s − 1.08i·17-s + 0.634i·19-s + (0.487 − 0.872i)21-s − 1.19·23-s + (0.458 + 0.888i)27-s − 0.742i·29-s − 0.222i·31-s + (0.563 − 1.47i)33-s − 0.735i·37-s + (0.838 + 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.478992279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478992279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 - 0.618i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.381 + 2.61i)T \) |
| good | 11 | \( 1 + 5.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 2.76iT - 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 2.76iT - 43T^{2} \) |
| 47 | \( 1 - 1.23iT - 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 2.76iT - 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
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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
−8.841612550807836065094812707283, −8.225358878792161520591737769113, −7.70843330751506609357521539676, −6.71325923315995986573004017300, −5.83129179390813868410786616690, −4.79685047382575170733098535089, −3.70540781558967976421812729742, −3.46009564728296745861080591504, −2.11313530346208615138571601281, −0.78553964172285230910912820514,
1.61867641882287369188480944393, 2.19317248594608305945248739461, 3.34644857901335353595600823458, 4.21836982007241641040517984975, 5.18519541226933495188114156487, 6.27078725969971808768819344592, 6.91354751653096579825811034095, 7.80842892535700707414324534925, 8.579578674363347589819890428107, 8.939972851634769085298003452454
