| L(s) = 1 | + (−2.60 + 0.104i)2-s + (−0.278 − 0.960i)3-s + (4.76 − 0.384i)4-s + (−0.705 + 0.267i)5-s + (0.824 + 2.47i)6-s + (0.861 − 2.02i)7-s + (−7.19 + 0.873i)8-s + (−0.845 + 0.534i)9-s + (1.80 − 0.769i)10-s + (−2.17 + 3.44i)11-s + (−1.69 − 4.47i)12-s + (0.939 + 3.48i)13-s + (−2.02 + 5.35i)14-s + (0.453 + 0.602i)15-s + (9.17 − 1.49i)16-s + (3.69 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−1.83 + 0.0741i)2-s + (−0.160 − 0.554i)3-s + (2.38 − 0.192i)4-s + (−0.315 + 0.119i)5-s + (0.336 + 1.00i)6-s + (0.325 − 0.764i)7-s + (−2.54 + 0.308i)8-s + (−0.281 + 0.178i)9-s + (0.571 − 0.243i)10-s + (−0.656 + 1.03i)11-s + (−0.489 − 1.29i)12-s + (0.260 + 0.965i)13-s + (−0.542 + 1.43i)14-s + (0.116 + 0.155i)15-s + (2.29 − 0.372i)16-s + (0.897 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.286522 + 0.212701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.286522 + 0.212701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 + 0.960i)T \) |
| 13 | \( 1 + (-0.939 - 3.48i)T \) |
| good | 2 | \( 1 + (2.60 - 0.104i)T + (1.99 - 0.160i)T^{2} \) |
| 5 | \( 1 + (0.705 - 0.267i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (-0.861 + 2.02i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (2.17 - 3.44i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (-3.69 - 1.57i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (6.05 + 3.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.272 - 0.472i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.236 + 5.87i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (4.18 - 4.72i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-9.47 - 1.93i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (1.15 - 0.335i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (-2.26 - 11.1i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (1.43 + 0.987i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (-0.291 - 2.39i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (2.22 - 13.7i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (-6.88 - 5.17i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.0808 + 1.00i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (1.91 + 1.84i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-0.0218 - 0.0416i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (0.892 - 1.29i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-3.76 - 15.2i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (-3.51 + 2.02i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.831 + 0.679i)T + (19.4 - 95.0i)T^{2} \) |
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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
−10.95050122033247706746177468424, −10.12067323932637800634222437282, −9.323125252101898991002585212278, −8.265404120400687680104907341618, −7.59050229918738015007319264430, −7.03491293161663536455752890298, −6.07723978202885098549517988773, −4.33782357665206577634866102588, −2.43516643908312921141918954471, −1.30844576907980996701709808853,
0.41484679488921540750620860702, 2.26300925391248700239638293886, 3.50711413557166654870028196048, 5.44184857368248352986445424905, 6.17559617228061062390173026714, 7.67688647775403049386864769266, 8.260312097722617126751486576596, 8.836212912530695374366357660400, 9.855405867590155403832596215408, 10.59948950281341724026918682262
