| L(s) = 1 | + (−0.132 − 0.920i)2-s + (0.209 − 0.459i)3-s + (6.84 − 2.01i)4-s + (6.87 − 7.93i)5-s + (−0.450 − 0.132i)6-s + (−25.4 + 16.3i)7-s + (−5.84 − 12.7i)8-s + (17.5 + 20.2i)9-s + (−8.20 − 5.27i)10-s + (−5.23 + 36.4i)11-s + (0.513 − 3.56i)12-s + (−54.9 − 35.2i)13-s + (18.4 + 21.2i)14-s + (−2.20 − 4.82i)15-s + (37.0 − 23.7i)16-s + (35.1 + 10.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0467 − 0.325i)2-s + (0.0403 − 0.0884i)3-s + (0.855 − 0.251i)4-s + (0.614 − 0.709i)5-s + (−0.0306 − 0.00900i)6-s + (−1.37 + 0.883i)7-s + (−0.258 − 0.565i)8-s + (0.648 + 0.748i)9-s + (−0.259 − 0.166i)10-s + (−0.143 + 0.997i)11-s + (0.0123 − 0.0858i)12-s + (−1.17 − 0.752i)13-s + (0.351 + 0.405i)14-s + (−0.0379 − 0.0830i)15-s + (0.578 − 0.371i)16-s + (0.502 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.19458 - 0.374991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19458 - 0.374991i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-92.8 + 59.6i)T \) |
| good | 2 | \( 1 + (0.132 + 0.920i)T + (-7.67 + 2.25i)T^{2} \) |
| 3 | \( 1 + (-0.209 + 0.459i)T + (-17.6 - 20.4i)T^{2} \) |
| 5 | \( 1 + (-6.87 + 7.93i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (25.4 - 16.3i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (5.23 - 36.4i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (54.9 + 35.2i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-35.1 - 10.3i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (40.4 - 11.8i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (55.2 + 16.2i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (24.3 + 53.2i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (189. + 218. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-229. + 264. i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (77.4 - 169. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 - 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-122. + 78.6i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (673. + 432. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-248. - 544. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-59.0 - 410. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-135. - 939. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-400. + 117. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (69.6 + 44.7i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (46.5 + 53.6i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-306. + 670. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (965. - 1.11e3i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−17.09475959858448150402026432736, −16.00132666537952304047548881226, −15.00400099348394574896815806545, −12.74034819779283832167285702433, −12.55294357917692906104868682046, −10.33791893080062741340017053285, −9.461934283887595818736800319842, −7.18793154044481722199526365263, −5.50225252904909989556169700459, −2.35620956707911008639467413910,
3.18412949618614137869641572474, 6.37502212666635071200245692288, 7.15417709464807085520527334839, 9.560992704053626452102119244343, 10.69723926986062841185047218014, 12.36716026658677566206413658580, 13.79900120559507281897888609981, 15.12226039282501804748828721289, 16.37391175692443431097891199136, 17.13635725121394020076650071436
