| L(s) = 1 | + (−1.80 − 0.206i)2-s + (−1.29 − 0.938i)3-s + (1.25 + 0.291i)4-s + (−0.564 + 0.825i)5-s + (2.13 + 1.95i)6-s + (−0.0685 − 0.0704i)7-s + (1.22 + 0.435i)8-s + (−0.139 − 0.430i)9-s + (1.18 − 1.36i)10-s + (−0.376 − 3.29i)11-s + (−1.34 − 1.55i)12-s + (−2.20 + 3.66i)13-s + (0.108 + 0.141i)14-s + (1.50 − 0.536i)15-s + (−4.41 − 2.17i)16-s + (−3.22 + 2.63i)17-s + ⋯ |
| L(s) = 1 | + (−1.27 − 0.146i)2-s + (−0.745 − 0.541i)3-s + (0.625 + 0.145i)4-s + (−0.252 + 0.369i)5-s + (0.870 + 0.798i)6-s + (−0.0258 − 0.0266i)7-s + (0.431 + 0.153i)8-s + (−0.0466 − 0.143i)9-s + (0.375 − 0.433i)10-s + (−0.113 − 0.993i)11-s + (−0.387 − 0.447i)12-s + (−0.612 + 1.01i)13-s + (0.0290 + 0.0377i)14-s + (0.388 − 0.138i)15-s + (−1.10 − 0.542i)16-s + (−0.782 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.356174 + 0.0642786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.356174 + 0.0642786i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.376 + 3.29i)T \) |
| good | 2 | \( 1 + (1.80 + 0.206i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (1.29 + 0.938i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (0.0685 + 0.0704i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (2.20 - 3.66i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (3.22 - 2.63i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (0.263 + 0.0150i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.379 + 2.64i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (0.892 - 2.29i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (-5.76 - 2.43i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.824 - 9.60i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-5.23 + 2.95i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-4.86 + 1.42i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-2.13 + 10.5i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-6.05 + 2.97i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-4.92 - 2.77i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (-7.14 + 0.819i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (6.40 - 14.0i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.93 - 11.1i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-4.39 - 8.33i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.439 - 15.3i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (-14.7 - 2.54i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (8.67 + 5.57i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.94 - 7.23i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.66707150517758915685016656275, −9.910350089868705231183197109449, −8.781665327792236330322442707614, −8.355624296159524976529772421762, −6.99064602721341410491655448330, −6.69175092552802299730366734881, −5.39901236696133855797729731299, −4.05020379801935807576861641827, −2.37655667999973907388542376992, −0.883994965465621572936736766774,
0.47569530126626548410980752092, 2.33597877226171935750180602198, 4.33027504419812437752761459059, 4.96153161557235255000241988281, 6.14647369263205383394858201995, 7.56749170290461224714885916883, 7.77451871547030139340293563929, 9.144598743482442541589963167631, 9.621194410050701589608409388083, 10.52187162356021034792500019769
