L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−1.03 + 1.98i)5-s + 1.00i·6-s + (0.181 − 0.181i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (0.669 + 2.13i)10-s + 2.46i·11-s + (0.707 + 0.707i)12-s + (−2.77 + 2.77i)13-s − 0.256i·14-s + (−0.669 − 2.13i)15-s − 1.00·16-s + (−3.68 − 3.68i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (−0.463 + 0.886i)5-s + 0.408i·6-s + (0.0684 − 0.0684i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.211 + 0.674i)10-s + 0.744i·11-s + (0.204 + 0.204i)12-s + (−0.769 + 0.769i)13-s − 0.0684i·14-s + (−0.172 − 0.550i)15-s − 0.250·16-s + (−0.894 − 0.894i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.109212 + 0.394884i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.109212 + 0.394884i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.03 - 1.98i)T \) |
| 31 | \( 1 + (4.89 - 2.64i)T \) |
good | 7 | \( 1 + (-0.181 + 0.181i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.46iT - 11T^{2} \) |
| 13 | \( 1 + (2.77 - 2.77i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.68 + 3.68i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.69iT - 19T^{2} \) |
| 23 | \( 1 + (4.12 - 4.12i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 37 | \( 1 + (-1.14 - 1.14i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.13T + 41T^{2} \) |
| 43 | \( 1 + (-0.169 + 0.169i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.17 - 4.17i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.45 - 8.45i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.83iT - 59T^{2} \) |
| 61 | \( 1 + 8.97iT - 61T^{2} \) |
| 67 | \( 1 + (-2.21 + 2.21i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + (-3.31 + 3.31i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.78T + 79T^{2} \) |
| 83 | \( 1 + (-7.11 + 7.11i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 + (8.14 - 8.14i)T - 97iT^{2} \) |
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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
−10.68925965394884765155378207727, −9.587497753866546950870102197609, −9.239735704522374915534867082275, −7.54674912236812921238024775261, −7.01057393978802959079882136344, −6.07467598527828304170235566059, −4.80124787951490467920114000176, −4.32227017199597871079911771503, −3.11591541010439288116986570366, −2.08804131605620435535293403690,
0.15981355023493446744635263398, 1.97189925501171005850719616569, 3.59593409837280703920378399021, 4.46443508876162569903905118105, 5.52870647100332733066158909861, 6.01872377247718681487655044031, 7.17692707801717021371943756226, 8.130930188799710775927004621480, 8.414547684264885545294848298331, 9.656393890673284194690615973424