| L(s) = 1 | + (−1.81 − 2.16i)2-s + (−3.47 + 9.55i)3-s + (−1.38 + 7.87i)4-s + (−6.10 − 34.6i)5-s + (27.0 − 9.83i)6-s + (33.4 − 57.9i)7-s + (19.5 − 11.3i)8-s + (−17.0 − 14.3i)9-s + (−63.8 + 76.1i)10-s + (−92.1 − 159. i)11-s + (−70.4 − 40.6i)12-s + (−4.54 − 12.4i)13-s + (−186. + 32.8i)14-s + (351. + 62.0i)15-s + (−60.1 − 21.8i)16-s + (−183. + 154. i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (−0.386 + 1.06i)3-s + (−0.0868 + 0.492i)4-s + (−0.244 − 1.38i)5-s + (0.750 − 0.273i)6-s + (0.683 − 1.18i)7-s + (0.306 − 0.176i)8-s + (−0.210 − 0.176i)9-s + (−0.638 + 0.761i)10-s + (−0.761 − 1.31i)11-s + (−0.488 − 0.282i)12-s + (−0.0269 − 0.0739i)13-s + (−0.951 + 0.167i)14-s + (1.56 + 0.275i)15-s + (−0.234 − 0.0855i)16-s + (−0.635 + 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.554772 - 0.646360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554772 - 0.646360i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.81 + 2.16i)T \) |
| 19 | \( 1 + (-356. - 54.5i)T \) |
| good | 3 | \( 1 + (3.47 - 9.55i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (6.10 + 34.6i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (-33.4 + 57.9i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (92.1 + 159. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (4.54 + 12.4i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (183. - 154. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-152. + 866. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (361. - 430. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (346. + 199. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 814. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-536. + 1.47e3i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (23.4 + 132. i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-3.04e3 - 2.55e3i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-3.06e3 - 540. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (3.71e3 + 4.42e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (284. - 1.61e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-25.5 + 30.4i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-8.60e3 + 1.51e3i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-7.36e3 - 2.68e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (436. - 1.19e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-94.5 + 163. i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.08e3 + 2.97e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-3.02e3 - 3.61e3i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
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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
−15.74390947236144104125605385726, −13.84766421733832884061658402058, −12.68941007223549536881948755891, −11.10766811949714007059630099210, −10.51272456661122496812264435713, −9.005611251907755245600024828462, −7.913289985318867266932323687152, −5.12298232961290969479692314978, −4.03337237995828809148562607965, −0.72834430742891061933968991603,
2.18746401169087867775577134336, 5.50219477566870126410856657615, 7.03325385084384462390430608378, 7.63297576351786822063087423665, 9.536060053953413760338592803645, 11.15727974587721059506028924038, 12.09437143969335821120984711273, 13.61558118918761509662622655061, 15.07888587757241187062813222538, 15.47914004578297252159798158223
