| L(s) = 1 | + (1.39 + 0.224i)2-s + (−1.37 + 2.94i)3-s + (1.89 + 0.626i)4-s + (1.97 − 1.38i)5-s + (−2.57 + 3.80i)6-s + (1.46 + 2.54i)7-s + (2.51 + 1.30i)8-s + (−4.85 − 5.78i)9-s + (3.06 − 1.48i)10-s + (−0.908 − 3.39i)11-s + (−4.45 + 4.73i)12-s + (0.958 + 2.05i)13-s + (1.48 + 3.88i)14-s + (1.35 + 7.70i)15-s + (3.21 + 2.38i)16-s + (−5.21 − 4.37i)17-s + ⋯ |
| L(s) = 1 | + (0.987 + 0.158i)2-s + (−0.792 + 1.69i)3-s + (0.949 + 0.313i)4-s + (0.882 − 0.617i)5-s + (−1.05 + 1.55i)6-s + (0.555 + 0.962i)7-s + (0.887 + 0.460i)8-s + (−1.61 − 1.92i)9-s + (0.969 − 0.469i)10-s + (−0.273 − 1.02i)11-s + (−1.28 + 1.36i)12-s + (0.265 + 0.570i)13-s + (0.395 + 1.03i)14-s + (0.350 + 1.98i)15-s + (0.803 + 0.595i)16-s + (−1.26 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0418 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44782 + 1.50972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44782 + 1.50972i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.224i)T \) |
| 19 | \( 1 + (2.72 - 3.40i)T \) |
| good | 3 | \( 1 + (1.37 - 2.94i)T + (-1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (-1.97 + 1.38i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 2.54i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.908 + 3.39i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.958 - 2.05i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (5.21 + 4.37i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.997 + 5.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.43 - 0.213i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (0.381 + 0.661i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.50 + 5.50i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.138 + 0.0502i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.92 - 4.17i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (2.52 + 3.01i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.675 - 0.472i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-0.710 - 8.11i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-4.22 - 2.95i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 12.3i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (9.87 + 1.74i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 5.44i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (7.90 - 2.87i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.248 + 0.926i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (9.66 + 3.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.67 + 1.99i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.72466909902407575749362013226, −11.20039467966695562327295315061, −10.34784091847651046420134785690, −9.149353087702754299591810346987, −8.538306834802686331547613055759, −6.31078949588316454669307267389, −5.72433587351066703064197985707, −4.93344789136526162558891007942, −4.15297470534312307080404865898, −2.54669788672368161066537520231,
1.54333177684052986418290640416, 2.45161679911073804799030044102, 4.51983446544240911397368241638, 5.71723898693605852522871740103, 6.55387242739820074451496472319, 7.13280292749810397986249820366, 8.060043713434938588271255349735, 10.21221081883004459716999319801, 10.89373741266046170682736298854, 11.54348327801089838795408078316
