| L(s) = 1 | + (0.347 − 1.96i)2-s + (−3.75 − 1.36i)4-s + (−0.657 + 0.551i)5-s + (−20.5 + 7.47i)7-s + (−4 + 6.92i)8-s + (0.857 + 1.48i)10-s + (37.2 + 31.2i)11-s + (9.56 + 54.2i)13-s + (7.58 + 43.0i)14-s + (12.2 + 10.2i)16-s + (−39.6 − 68.5i)17-s + (−55.0 + 95.2i)19-s + (3.22 − 1.17i)20-s + (74.4 − 62.4i)22-s + (42.5 + 15.4i)23-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.469 − 0.171i)4-s + (−0.0587 + 0.0493i)5-s + (−1.10 + 0.403i)7-s + (−0.176 + 0.306i)8-s + (0.0271 + 0.0469i)10-s + (1.01 + 0.855i)11-s + (0.203 + 1.15i)13-s + (0.144 + 0.821i)14-s + (0.191 + 0.160i)16-s + (−0.565 − 0.978i)17-s + (−0.664 + 1.15i)19-s + (0.0360 − 0.0131i)20-s + (0.721 − 0.605i)22-s + (0.385 + 0.140i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.870385 + 0.537459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.870385 + 0.537459i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.657 - 0.551i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (20.5 - 7.47i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-37.2 - 31.2i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-9.56 - 54.2i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (39.6 + 68.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (55.0 - 95.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-42.5 - 15.4i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (21.6 - 123. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-125. - 45.7i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (84.1 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-3.71 - 21.0i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (128. + 108. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-298. + 108. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (415. - 348. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (648. - 236. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-111. - 633. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (524. + 908. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-480. + 831. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-39.7 + 225. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-29.5 + 167. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (379. - 657. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.23e3 - 1.03e3i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−12.34709182187270381674898457801, −11.81817653343032258449424927910, −10.60317491712595988680677173913, −9.388695761178152552813488087272, −9.045864198971638969475059947959, −7.14572571245088653541563946673, −6.17889527990230240314517144432, −4.55170747343101461565694571901, −3.37701064326012128012958874598, −1.79023271585638715351906430780,
0.46339038851669463170084809178, 3.18371520942227038298001926525, 4.38542960258398761827654428342, 6.11183990327696437244795076219, 6.59698734739715826733474949639, 8.082648395738989823686531898342, 8.971789430260613980408274308331, 10.10299775083751074775526329961, 11.16631916493384931214958950147, 12.57499756074018151043408712127
