| L(s) = 1 | + (−0.443 − 1.34i)2-s + (2.49 − 1.16i)3-s + (−1.60 + 1.19i)4-s + (−0.636 + 0.909i)5-s + (−2.66 − 2.83i)6-s + (1.63 − 2.83i)7-s + (2.31 + 1.62i)8-s + (2.93 − 3.49i)9-s + (1.50 + 0.451i)10-s + (−5.36 − 1.43i)11-s + (−2.61 + 4.83i)12-s + (2.49 + 1.16i)13-s + (−4.52 − 0.939i)14-s + (−0.530 + 3.00i)15-s + (1.16 − 3.82i)16-s + (5.30 − 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.313 − 0.949i)2-s + (1.43 − 0.670i)3-s + (−0.803 + 0.595i)4-s + (−0.284 + 0.406i)5-s + (−1.08 − 1.15i)6-s + (0.618 − 1.07i)7-s + (0.817 + 0.575i)8-s + (0.976 − 1.16i)9-s + (0.475 + 0.142i)10-s + (−1.61 − 0.433i)11-s + (−0.755 + 1.39i)12-s + (0.692 + 0.322i)13-s + (−1.21 − 0.251i)14-s + (−0.136 + 0.776i)15-s + (0.290 − 0.956i)16-s + (1.28 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.894701 - 1.29745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.894701 - 1.29745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.443 + 1.34i)T \) |
| 19 | \( 1 + (2.08 + 3.82i)T \) |
| good | 3 | \( 1 + (-2.49 + 1.16i)T + (1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (0.636 - 0.909i)T + (-1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 2.83i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.36 + 1.43i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.49 - 1.16i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-5.30 + 4.45i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.991 - 5.62i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.783 - 8.95i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-0.847 + 1.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 - 1.62i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.15 + 0.786i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.17 - 2.22i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (3.48 - 4.15i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-6.03 - 8.62i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (9.33 + 0.816i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (1.77 + 2.53i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-3.19 + 0.279i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-3.83 + 0.676i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.36 + 6.48i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (3.84 + 1.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.66 + 2.05i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (2.67 - 0.974i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.37 - 7.59i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.18788971110126731967189288529, −10.64954843078505090301761171563, −9.496094504127243826532632379497, −8.548624322875091958003305126218, −7.61185990533218110448723122701, −7.37914855309912289539071485197, −5.00051976440175814272466518939, −3.50551037818507893761592819493, −2.85031623598978656989759809775, −1.31580190589579586481877556905,
2.26690436576070388140906147341, 3.88845167000302649367899561934, 4.98550573733715740050399852083, 5.97686454708306344330574746007, 7.941951017735054046519731992389, 8.167708805423722665697685786209, 8.706554583606620879491458820755, 10.03060497217196428424481094943, 10.42326271212060327828430560598, 12.31593433977005956383941925718
