| L(s) = 1 | + (1.42 − 8.09i)3-s + (−6.74 − 5.66i)5-s + (0.266 + 0.461i)7-s + (−38.1 − 13.8i)9-s + (−18.3 + 31.8i)11-s + (−8.11 − 46.0i)13-s + (−55.4 + 46.5i)15-s + (92.4 − 33.6i)17-s + (−51.1 − 65.1i)19-s + (4.11 − 1.49i)21-s + (81.4 − 68.3i)23-s + (−8.22 − 46.6i)25-s + (−55.7 + 96.4i)27-s + (81.7 + 29.7i)29-s + (121. + 210. i)31-s + ⋯ |
| L(s) = 1 | + (0.274 − 1.55i)3-s + (−0.603 − 0.506i)5-s + (0.0143 + 0.0248i)7-s + (−1.41 − 0.513i)9-s + (−0.503 + 0.871i)11-s + (−0.173 − 0.981i)13-s + (−0.954 + 0.801i)15-s + (1.31 − 0.480i)17-s + (−0.617 − 0.786i)19-s + (0.0427 − 0.0155i)21-s + (0.738 − 0.619i)23-s + (−0.0658 − 0.373i)25-s + (−0.397 + 0.687i)27-s + (0.523 + 0.190i)29-s + (0.703 + 1.21i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.434263 - 1.18198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.434263 - 1.18198i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (51.1 + 65.1i)T \) |
| good | 3 | \( 1 + (-1.42 + 8.09i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (6.74 + 5.66i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-0.266 - 0.461i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (18.3 - 31.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.11 + 46.0i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-92.4 + 33.6i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-81.4 + 68.3i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-81.7 - 29.7i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-121. - 210. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (56.4 - 319. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (217. + 182. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-231. - 84.1i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-363. + 304. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (500. - 181. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-418. + 351. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (885. + 322. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-355. - 298. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (81.3 - 461. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (139. - 791. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-466. - 808. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (192. + 1.09e3i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-899. + 327. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.28999389339038272506430524370, −12.54020833475436572870309614195, −11.92488648870683066682965616727, −10.26559295436817120479895983034, −8.535628188134328229669881366518, −7.74824454870359409347471281989, −6.75856391352290296939070743962, −5.01848966330496471774375600478, −2.72663037044752166615128313474, −0.811854002885409289921763328445,
3.20485192077724889183843383789, 4.24727816632983686966485572247, 5.78412280914362823823102699504, 7.70754431401760788672419753422, 8.921520973508521780205575748835, 10.08046253541768226444423356715, 10.90190624283364671177194392827, 11.92192501775819841580719166923, 13.69208052096958775834568270657, 14.74944395848797958853464395563
