| L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.27 − 1.17i)3-s + (−0.959 + 0.281i)4-s + (2.23 − 0.136i)5-s + (−1.34 − 1.09i)6-s + (−1.96 + 1.26i)7-s + (0.415 + 0.909i)8-s + (0.250 − 2.98i)9-s + (−0.452 − 2.18i)10-s + (−0.590 + 4.10i)11-s + (−0.892 + 1.48i)12-s + (2.60 − 4.05i)13-s + (1.53 + 1.76i)14-s + (2.68 − 2.79i)15-s + (0.841 − 0.540i)16-s + (2.07 − 7.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.736 − 0.676i)3-s + (−0.479 + 0.140i)4-s + (0.998 − 0.0611i)5-s + (−0.547 − 0.447i)6-s + (−0.743 + 0.477i)7-s + (0.146 + 0.321i)8-s + (0.0835 − 0.996i)9-s + (−0.143 − 0.692i)10-s + (−0.178 + 1.23i)11-s + (−0.257 + 0.428i)12-s + (0.722 − 1.12i)13-s + (0.409 + 0.472i)14-s + (0.693 − 0.720i)15-s + (0.210 − 0.135i)16-s + (0.503 − 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13076 - 1.53922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13076 - 1.53922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-1.27 + 1.17i)T \) |
| 5 | \( 1 + (-2.23 + 0.136i)T \) |
| 23 | \( 1 + (-0.585 + 4.75i)T \) |
| good | 7 | \( 1 + (1.96 - 1.26i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.590 - 4.10i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 4.05i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.07 + 7.06i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.43 + 4.89i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.892 - 3.03i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.520 - 1.14i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (1.29 + 1.49i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-8.22 - 7.12i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.57 - 5.64i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + (-7.10 - 11.0i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.99 - 3.10i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.51 - 2.51i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.160 - 1.11i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.381 + 0.0548i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.69 + 15.9i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.72 - 5.79i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (12.0 - 10.4i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.45 - 14.1i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.68 - 6.56i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.01563464716993162866069514833, −9.328559167751027093441267525898, −8.858150274198853377411958832396, −7.63124276847980118924927168522, −6.76752152440365871771588052687, −5.74442848865733118873272981537, −4.59901943583386146791324320827, −2.84139955869213908510189261546, −2.63965460464905853115526625345, −1.05458034777757828458312647210,
1.76092351941604125381628628835, 3.46021384000038498491628725702, 4.04792931201355516463724589355, 5.69834555750920888873100830738, 6.04664452472938643283996997361, 7.23452570819299549835983611044, 8.428024121605610366896660483871, 8.833669378920682265333194002218, 9.881964659895099298235951625157, 10.29669870329700226437851991829
