| L(s) = 1 | + (2.25 − 1.44i)2-s + (−0.677 + 4.71i)3-s + (1.31 − 2.88i)4-s + (0.629 + 2.14i)5-s + (5.29 + 11.6i)6-s + (−1.25 + 1.08i)7-s + (0.316 + 2.20i)8-s + (−13.1 − 3.85i)9-s + (4.52 + 3.92i)10-s + (2.89 − 4.50i)11-s + (12.7 + 8.16i)12-s + (7.81 − 9.01i)13-s + (−1.25 + 4.26i)14-s + (−10.5 + 1.51i)15-s + (12.2 + 14.0i)16-s + (29.5 − 13.5i)17-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.723i)2-s + (−0.225 + 1.57i)3-s + (0.329 − 0.720i)4-s + (0.125 + 0.429i)5-s + (0.883 + 1.93i)6-s + (−0.179 + 0.155i)7-s + (0.0395 + 0.275i)8-s + (−1.45 − 0.428i)9-s + (0.452 + 0.392i)10-s + (0.263 − 0.409i)11-s + (1.05 + 0.680i)12-s + (0.600 − 0.693i)13-s + (−0.0893 + 0.304i)14-s + (−0.702 + 0.101i)15-s + (0.762 + 0.880i)16-s + (1.74 − 0.795i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.04623 + 0.838689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04623 + 0.838689i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.629 - 2.14i)T \) |
| 23 | \( 1 + (19.3 + 12.4i)T \) |
| good | 2 | \( 1 + (-2.25 + 1.44i)T + (1.66 - 3.63i)T^{2} \) |
| 3 | \( 1 + (0.677 - 4.71i)T + (-8.63 - 2.53i)T^{2} \) |
| 7 | \( 1 + (1.25 - 1.08i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (-2.89 + 4.50i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (-7.81 + 9.01i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (-29.5 + 13.5i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (14.1 + 6.47i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (-5.69 - 12.4i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (4.60 + 32.0i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (5.34 - 18.2i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (30.1 - 8.86i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (-7.46 - 1.07i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 + 35.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-43.8 + 38.0i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (3.98 - 4.60i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-69.6 + 10.0i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (50.5 + 78.7i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (89.6 - 57.6i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (-49.9 + 109. i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (-111. - 97.0i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (8.47 - 28.8i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (100. + 14.4i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (8.47 + 28.8i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
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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
−13.54219967801005174990930890584, −12.27482788813156202343659588865, −11.33296587584591276467000490559, −10.53777259350903981286394719491, −9.715921104727027200125447528650, −8.276120000234588266433515363175, −6.02241629786693824206277437781, −5.10573581863923056221963082048, −3.85581708000227518714735468025, −3.00178955900365115090508151829,
1.48070664175476133638167612542, 3.86289674474087093013022980241, 5.55222213452455966621289235047, 6.35200699859530971632798139514, 7.28124049317274375726573612736, 8.346375446195974983118889580199, 10.07242364793625496765772611476, 11.91809144821132272756950539659, 12.43576985694099748806490890795, 13.27396497423290648431141221945
