L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.451 + 1.67i)3-s − 1.00i·4-s + (0.664 + 2.13i)5-s + (−1.50 − 0.862i)6-s + (2.34 + 2.34i)7-s + (0.707 + 0.707i)8-s + (−2.59 + 1.51i)9-s + (−1.97 − 1.03i)10-s + 4.92i·11-s + (1.67 − 0.451i)12-s + (2.78 − 2.78i)13-s − 3.31·14-s + (−3.26 + 2.07i)15-s − 1.00·16-s + (2.05 − 2.05i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.260 + 0.965i)3-s − 0.500i·4-s + (0.297 + 0.954i)5-s + (−0.613 − 0.352i)6-s + (0.884 + 0.884i)7-s + (0.250 + 0.250i)8-s + (−0.863 + 0.503i)9-s + (−0.625 − 0.328i)10-s + 1.48i·11-s + (0.482 − 0.130i)12-s + (0.772 − 0.772i)13-s − 0.884·14-s + (−0.844 + 0.536i)15-s − 0.250·16-s + (0.497 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.324606 + 1.41340i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.324606 + 1.41340i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.451 - 1.67i)T \) |
| 5 | \( 1 + (-0.664 - 2.13i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-2.34 - 2.34i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.92iT - 11T^{2} \) |
| 13 | \( 1 + (-2.78 + 2.78i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.05 + 2.05i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.73iT - 19T^{2} \) |
| 29 | \( 1 - 9.53T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 + (1.67 + 1.67i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.18iT - 41T^{2} \) |
| 43 | \( 1 + (-2.73 + 2.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.67 - 8.67i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.92 + 8.92i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.44T + 59T^{2} \) |
| 61 | \( 1 + 2.83T + 61T^{2} \) |
| 67 | \( 1 + (6.60 + 6.60i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.39iT - 71T^{2} \) |
| 73 | \( 1 + (-7.65 + 7.65i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.62iT - 79T^{2} \) |
| 83 | \( 1 + (0.357 + 0.357i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 + (-5.52 - 5.52i)T + 97iT^{2} \) |
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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
−10.65197217286428031908403775245, −9.885169137269582791053896343306, −9.201880552833547562661338637551, −8.293688952178172885936244625801, −7.51330770719259594717016674880, −6.42495817791738172583967909926, −5.36133123339724617234203576697, −4.69615735169674590131131066419, −3.13866975288570138025366259931, −2.07062208244596513657105089566,
0.988328550153633662540079317673, 1.59762016135038809888338679595, 3.25436254215923092543334961762, 4.36880882266678424918756891755, 5.74358291928791700659899276474, 6.62797736551164182443517470247, 7.976729821904154849914447628454, 8.285732710187307227048486038813, 8.958652795077989869359659714219, 10.13868637416428281347516770803