| L(s) = 1 | + (1.25 + 0.143i)2-s + (0.302 + 0.219i)3-s + (−0.401 − 0.0933i)4-s + (−0.564 + 0.825i)5-s + (0.347 + 0.318i)6-s + (−2.98 − 3.07i)7-s + (−2.86 − 1.02i)8-s + (−0.883 − 2.72i)9-s + (−0.825 + 0.952i)10-s + (2.93 − 1.55i)11-s + (−0.100 − 0.116i)12-s + (−2.19 + 3.64i)13-s + (−3.29 − 4.27i)14-s + (−0.352 + 0.125i)15-s + (−2.69 − 1.32i)16-s + (4.31 − 3.52i)17-s + ⋯ |
| L(s) = 1 | + (0.885 + 0.101i)2-s + (0.174 + 0.126i)3-s + (−0.200 − 0.0466i)4-s + (−0.252 + 0.369i)5-s + (0.141 + 0.130i)6-s + (−1.12 − 1.16i)7-s + (−1.01 − 0.361i)8-s + (−0.294 − 0.906i)9-s + (−0.260 + 0.301i)10-s + (0.883 − 0.467i)11-s + (−0.0291 − 0.0335i)12-s + (−0.609 + 1.01i)13-s + (−0.881 − 1.14i)14-s + (−0.0909 + 0.0324i)15-s + (−0.674 − 0.331i)16-s + (1.04 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.554307 - 0.901353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554307 - 0.901353i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (-2.93 + 1.55i)T \) |
| good | 2 | \( 1 + (-1.25 - 0.143i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-0.302 - 0.219i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (2.98 + 3.07i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (2.19 - 3.64i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-4.31 + 3.52i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (6.44 + 0.368i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.376 + 2.61i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (0.678 - 1.74i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (8.04 + 3.39i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.534 - 6.22i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-7.43 + 4.20i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-8.97 + 2.63i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (1.20 - 5.96i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-6.58 + 3.23i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (5.44 + 3.07i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (2.29 - 0.263i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (0.627 - 1.37i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.738 - 2.81i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (7.58 + 14.3i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (0.0204 + 0.714i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (4.14 + 0.717i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (14.4 + 9.31i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-8.95 - 13.0i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.30279737371095972597035417405, −9.373850467905266055375731853004, −8.985331245024300012988399902822, −7.33247368731164671803718788905, −6.59323655632905962925454613401, −5.95431049073028091227436904878, −4.35826937038905408341663937823, −3.83631923390615875885106603426, −2.99435372414963491429824392934, −0.41886466907620485705158376254,
2.29112235250588300583897030333, 3.33261467848348489110980527256, 4.32374182827663344777884109293, 5.59511383398848724986735534093, 5.89493100771347779866169516017, 7.38770425554201242867904989069, 8.467439248059953567747779034421, 9.084977519231488313888022926054, 9.965362104342417442474081100854, 11.12694219446251554465772727522
