L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.53 + 0.805i)3-s − 1.00i·4-s + (1.53 + 1.62i)5-s + (−0.514 + 1.65i)6-s + (−0.677 − 0.677i)7-s + (−0.707 − 0.707i)8-s + (1.70 − 2.47i)9-s + (2.23 + 0.0614i)10-s + 2.37·11-s + (0.805 + 1.53i)12-s + (0.799 + 3.51i)13-s − 0.957·14-s + (−3.66 − 1.25i)15-s − 1.00·16-s + (5.31 + 5.31i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.885 + 0.465i)3-s − 0.500i·4-s + (0.687 + 0.726i)5-s + (−0.209 + 0.675i)6-s + (−0.255 − 0.255i)7-s + (−0.250 − 0.250i)8-s + (0.567 − 0.823i)9-s + (0.706 + 0.0194i)10-s + 0.716·11-s + (0.232 + 0.442i)12-s + (0.221 + 0.975i)13-s − 0.255·14-s + (−0.946 − 0.323i)15-s − 0.250·16-s + (1.29 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54102 + 0.0975227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54102 + 0.0975227i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.53 - 0.805i)T \) |
| 5 | \( 1 + (-1.53 - 1.62i)T \) |
| 13 | \( 1 + (-0.799 - 3.51i)T \) |
good | 7 | \( 1 + (0.677 + 0.677i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 + (-5.31 - 5.31i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 + (-1.90 + 1.90i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 8.49iT - 31T^{2} \) |
| 37 | \( 1 + (2.64 + 2.64i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.75T + 41T^{2} \) |
| 43 | \( 1 + (-0.132 - 0.132i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.08 + 2.08i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.82 - 5.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 + (8.47 + 8.47i)T + 67iT^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + (3.69 - 3.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.03iT - 79T^{2} \) |
| 83 | \( 1 + (0.475 + 0.475i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.15iT - 89T^{2} \) |
| 97 | \( 1 + (-1.11 - 1.11i)T + 97iT^{2} \) |
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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
−11.38933501426344505727338575770, −10.39339448049577576319348445899, −9.961594743077418201512994682375, −8.993236866997444690760380480516, −7.15782916286309961116359753220, −6.27662231822076129145393855320, −5.63539830382039676085284386855, −4.26570191990943832672697263426, −3.37042895769667088630981625004, −1.54904848447065725290225087046,
1.20614492226452982951392986495, 3.16298929631201187123691827257, 4.95349365375785204946118102829, 5.41425813998493431643568873685, 6.36949011925352923203190482583, 7.29722103517805323948819195832, 8.371324163973698046194838713563, 9.496078759387427979687797963692, 10.37157066174039131099134012724, 11.72764295329896885182296792816