| L(s) = 1 | + (−0.281 − 0.959i)2-s + (0.951 − 0.824i)3-s + (−0.841 + 0.540i)4-s + (−1.90 − 1.16i)5-s + (−1.05 − 0.680i)6-s + (−4.71 − 2.15i)7-s + (0.755 + 0.654i)8-s + (−0.201 + 1.40i)9-s + (−0.582 + 2.15i)10-s + (−2.19 − 0.643i)11-s + (−0.354 + 1.20i)12-s + (2.34 − 1.06i)13-s + (−0.737 + 5.13i)14-s + (−2.77 + 0.462i)15-s + (0.415 − 0.909i)16-s + (3.46 − 5.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 − 0.678i)2-s + (0.549 − 0.475i)3-s + (−0.420 + 0.270i)4-s + (−0.853 − 0.521i)5-s + (−0.432 − 0.277i)6-s + (−1.78 − 0.814i)7-s + (0.267 + 0.231i)8-s + (−0.0671 + 0.467i)9-s + (−0.184 + 0.682i)10-s + (−0.660 − 0.193i)11-s + (−0.102 + 0.348i)12-s + (0.649 − 0.296i)13-s + (−0.197 + 1.37i)14-s + (−0.716 + 0.119i)15-s + (0.103 − 0.227i)16-s + (0.840 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0542254 - 0.697705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0542254 - 0.697705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.281 + 0.959i)T \) |
| 5 | \( 1 + (1.90 + 1.16i)T \) |
| 23 | \( 1 + (4.75 + 0.619i)T \) |
| good | 3 | \( 1 + (-0.951 + 0.824i)T + (0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (4.71 + 2.15i)T + (4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.19 + 0.643i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.34 + 1.06i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.46 + 5.39i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.04 + 3.24i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (1.24 + 0.797i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.0386 + 0.0446i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (2.82 + 0.405i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.69 + 11.7i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.17 - 1.88i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 2.51iT - 47T^{2} \) |
| 53 | \( 1 + (-8.37 - 3.82i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (4.59 + 10.0i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (1.22 - 1.41i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.16 - 3.96i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.50 + 0.736i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.81 - 5.94i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (2.52 + 5.52i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.603 - 0.0868i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.26 - 1.45i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-5.40 + 0.776i)T + (93.0 - 27.3i)T^{2} \) |
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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
−11.89425316327619808950090348489, −10.73288153674718116605991058345, −9.825504039663704458728189559376, −8.882580207806987351180271725561, −7.75261922736489557305593052740, −7.12703979511174712331543311880, −5.31623626338257851742728047446, −3.69703993750540836951235164894, −2.90869018086850885275089598401, −0.57449984001680741860656023686,
3.16848991142761873361640246806, 3.81538833811949342550842644235, 5.79796963107692322198253836426, 6.53615195475265387940515263974, 7.82524621931174094967768369798, 8.687481216245026326543028595960, 9.745371567928696743760844279848, 10.25786168019934416556305801819, 11.90369879933263085139191350377, 12.61645729188365528993881106325
