| L(s) = 1 | + (1.29 − 0.579i)2-s + (−0.756 + 1.82i)3-s + (1.32 − 1.49i)4-s + (−1.28 + 2.52i)5-s + (0.0819 + 2.79i)6-s + (0.634 + 2.64i)7-s + (0.848 − 2.69i)8-s + (−0.642 − 0.642i)9-s + (−0.197 + 4.00i)10-s + (3.28 − 3.84i)11-s + (1.72 + 3.55i)12-s + (−2.94 − 4.80i)13-s + (2.35 + 3.04i)14-s + (−3.63 − 4.25i)15-s + (−0.467 − 3.97i)16-s + (0.0571 + 0.726i)17-s + ⋯ |
| L(s) = 1 | + (0.912 − 0.409i)2-s + (−0.436 + 1.05i)3-s + (0.664 − 0.747i)4-s + (−0.575 + 1.12i)5-s + (0.0334 + 1.14i)6-s + (0.239 + 0.999i)7-s + (0.300 − 0.953i)8-s + (−0.214 − 0.214i)9-s + (−0.0623 + 1.26i)10-s + (0.989 − 1.15i)11-s + (0.497 + 1.02i)12-s + (−0.815 − 1.33i)13-s + (0.628 + 0.813i)14-s + (−0.939 − 1.09i)15-s + (−0.116 − 0.993i)16-s + (0.0138 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48300 + 0.558588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48300 + 0.558588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.579i)T \) |
| 41 | \( 1 + (-4.55 + 4.49i)T \) |
| good | 3 | \( 1 + (0.756 - 1.82i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (1.28 - 2.52i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.634 - 2.64i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-3.28 + 3.84i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.94 + 4.80i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.0571 - 0.726i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-0.531 - 0.325i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.882 - 0.641i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.452 - 5.75i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-2.23 + 6.88i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.79 + 5.52i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.642 + 4.05i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (2.91 - 12.1i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (7.02 + 0.553i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (2.32 - 3.20i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.330 - 2.08i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (2.50 - 2.13i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-9.35 - 7.99i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-3.14 + 3.14i)T - 73iT^{2} \) |
| 79 | \( 1 + (14.9 + 6.20i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 6.63iT - 83T^{2} \) |
| 89 | \( 1 + (-7.99 + 1.91i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (8.34 - 7.13i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.75470752236800135023714968018, −11.72355613673914208496243192646, −11.05967749625458218918802020777, −10.42050939785026944637024072405, −9.237428547390641911913720068100, −7.54597194176658117146610176999, −6.09201383532742697937021522805, −5.26443982677977395049133241680, −3.85618050864359523968938752972, −2.87234870246450912602656299311,
1.58632679217149090413186800423, 4.19475987191093343757000223401, 4.79086618163283825362939421945, 6.63303433872611415203405196350, 7.11995855028053412582967729499, 8.131285533957920576457530765510, 9.592671509204550412941647268157, 11.45183534318357363808283021698, 12.07887840409622054231561902350, 12.58497786093277939487807281318
