| L(s) = 1 | + (−1.40 − 0.121i)2-s + (0.708 + 1.55i)3-s + (1.97 + 0.342i)4-s + (0.680 + 0.785i)5-s + (−0.810 − 2.27i)6-s + (1.52 + 0.978i)7-s + (−2.73 − 0.722i)8-s + (0.0576 − 0.0665i)9-s + (−0.863 − 1.18i)10-s + (−2.78 + 0.400i)11-s + (0.864 + 3.30i)12-s + (1.06 + 1.66i)13-s + (−2.02 − 1.56i)14-s + (−0.736 + 1.61i)15-s + (3.76 + 1.35i)16-s + (−0.326 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0859i)2-s + (0.409 + 0.896i)3-s + (0.985 + 0.171i)4-s + (0.304 + 0.351i)5-s + (−0.330 − 0.928i)6-s + (0.575 + 0.369i)7-s + (−0.966 − 0.255i)8-s + (0.0192 − 0.0221i)9-s + (−0.273 − 0.376i)10-s + (−0.840 + 0.120i)11-s + (0.249 + 0.953i)12-s + (0.296 + 0.461i)13-s + (−0.541 − 0.418i)14-s + (−0.190 + 0.416i)15-s + (0.941 + 0.337i)16-s + (−0.0791 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.790902 + 0.516229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790902 + 0.516229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.121i)T \) |
| 23 | \( 1 + (-2.59 - 4.03i)T \) |
| good | 3 | \( 1 + (-0.708 - 1.55i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.680 - 0.785i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.52 - 0.978i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (2.78 - 0.400i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 1.66i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.326 + 1.11i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.13 - 3.86i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.543 + 1.85i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.29 + 1.50i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.398 + 0.460i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.94 + 5.70i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.20 + 1.46i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 + (-0.454 - 0.291i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.00 - 5.14i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.76 + 12.6i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-7.52 - 1.08i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-8.75 - 1.25i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.33 - 1.85i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.58 - 2.94i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.507 + 0.439i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.31 + 3.33i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (2.21 - 1.92i)T + (13.8 - 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55700403267791870914160472663, −11.45965540582407799767412393266, −10.53051436507776950807448121441, −9.831624487490309084646251895676, −8.918549559499523048609760629403, −8.018763214963181501758243965994, −6.77721165466533107314524885604, −5.36185889157516530811034214346, −3.65611200613206499275531210440, −2.16523256141015484501356362748,
1.29576914661183492010839906178, 2.69683366725616205396610700060, 5.07999656532175739976692894531, 6.54450414651520058140960269940, 7.57610237440975788435241154764, 8.241368988054733480833920734584, 9.192242395700548417026010105083, 10.53772944133644432921299984981, 11.13288583808368348928316850515, 12.63169954476927584857160450193
