| 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
−14.74944395848797958853464395563, −13.69208052096958775834568270657, −11.92192501775819841580719166923, −10.90190624283364671177194392827, −10.08046253541768226444423356715, −8.921520973508521780205575748835, −7.70754431401760788672419753422, −5.78412280914362823823102699504, −4.24727816632983686966485572247, −3.20485192077724889183843383789,
0.811854002885409289921763328445, 2.72663037044752166615128313474, 5.01848966330496471774375600478, 6.75856391352290296939070743962, 7.74824454870359409347471281989, 8.535628188134328229669881366518, 10.26559295436817120479895983034, 11.92488648870683066682965616727, 12.54020833475436572870309614195, 13.28999389339038272506430524370
