| L(s) = 1 | + (3.29 + 0.968i)3-s + (4.37 + 6.80i)5-s + (−9.96 − 1.43i)7-s + (2.36 + 1.52i)9-s + (10.2 − 4.66i)11-s + (−1.17 − 8.15i)13-s + (7.83 + 26.6i)15-s + (5.11 + 4.43i)17-s + (16.5 − 14.3i)19-s + (−31.4 − 14.3i)21-s + (−19.1 − 12.7i)23-s + (−16.7 + 36.7i)25-s + (−13.9 − 16.0i)27-s + (21.0 − 24.3i)29-s + (2.39 − 0.701i)31-s + ⋯ |
| L(s) = 1 | + (1.09 + 0.322i)3-s + (0.874 + 1.36i)5-s + (−1.42 − 0.204i)7-s + (0.262 + 0.168i)9-s + (0.928 − 0.424i)11-s + (−0.0901 − 0.626i)13-s + (0.522 + 1.77i)15-s + (0.300 + 0.260i)17-s + (0.871 − 0.755i)19-s + (−1.49 − 0.684i)21-s + (−0.833 − 0.552i)23-s + (−0.671 + 1.47i)25-s + (−0.515 − 0.595i)27-s + (0.727 − 0.839i)29-s + (0.0771 − 0.0226i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.69334 + 0.605461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69334 + 0.605461i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (19.1 + 12.7i)T \) |
| good | 3 | \( 1 + (-3.29 - 0.968i)T + (7.57 + 4.86i)T^{2} \) |
| 5 | \( 1 + (-4.37 - 6.80i)T + (-10.3 + 22.7i)T^{2} \) |
| 7 | \( 1 + (9.96 + 1.43i)T + (47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (-10.2 + 4.66i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (1.17 + 8.15i)T + (-162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (-5.11 - 4.43i)T + (41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (-16.5 + 14.3i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-21.0 + 24.3i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 0.701i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (38.0 - 59.2i)T + (-568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (52.3 - 33.6i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-10.5 + 35.8i)T + (-1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + 57.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-48.3 - 6.95i)T + (2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (-5.90 - 41.0i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-21.1 - 72.1i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-90.7 - 41.4i)T + (2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-8.90 + 19.5i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (21.4 + 24.7i)T + (-758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (69.9 - 10.0i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-27.3 + 42.6i)T + (-2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (0.564 - 1.92i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (72.0 + 112. i)T + (-3.90e3 + 8.55e3i)T^{2} \) |
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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
−13.88676958600111359501748188000, −13.42572614721567027458788959992, −11.76876211670710710710190653928, −10.09004980680115758543269612400, −9.879959002214354018909774731833, −8.560155584302475788955216271281, −6.92145599170966195487858215080, −6.08707544261333919041414068117, −3.50383040931256596684056078182, −2.78373174145778066648427557050,
1.78582710234218088521637559351, 3.57822442126549693293980648264, 5.45962530448040839802982060214, 6.84173643908510892283221565928, 8.409439005865469031966177251989, 9.390009577171025924349944187482, 9.729309279623302173733573779337, 12.08727587089294182894825529685, 12.75239031207825636031968281520, 13.76725476238688493182831352249
