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
−10.52187162356021034792500019769, −9.621194410050701589608409388083, −9.144598743482442541589963167631, −7.77451871547030139340293563929, −7.56749170290461224714885916883, −6.14647369263205383394858201995, −4.96153161557235255000241988281, −4.33027504419812437752761459059, −2.33597877226171935750180602198, −0.47569530126626548410980752092,
0.883994965465621572936736766774, 2.37655667999973907388542376992, 4.05020379801935807576861641827, 5.39901236696133855797729731299, 6.69175092552802299730366734881, 6.99064602721341410491655448330, 8.355624296159524976529772421762, 8.781665327792236330322442707614, 9.910350089868705231183197109449, 10.66707150517758915685016656275