L(s) = 1 | − 2-s + (1.44 + 0.951i)3-s + 4-s + (−1.42 + 1.71i)5-s + (−1.44 − 0.951i)6-s − 4.73·7-s − 8-s + (1.18 + 2.75i)9-s + (1.42 − 1.71i)10-s + 0.109·11-s + (1.44 + 0.951i)12-s − 2.46i·13-s + 4.73·14-s + (−3.70 + 1.12i)15-s + 16-s + 3.72i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.835 + 0.549i)3-s + 0.5·4-s + (−0.639 + 0.768i)5-s + (−0.590 − 0.388i)6-s − 1.79·7-s − 0.353·8-s + (0.395 + 0.918i)9-s + (0.452 − 0.543i)10-s + 0.0330·11-s + (0.417 + 0.274i)12-s − 0.685i·13-s + 1.26·14-s + (−0.956 + 0.290i)15-s + 0.250·16-s + 0.902i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0439275 - 0.166607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0439275 - 0.166607i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.44 - 0.951i)T \) |
| 5 | \( 1 + (1.42 - 1.71i)T \) |
| 23 | \( 1 + (4.67 - 1.04i)T \) |
good | 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.109T + 11T^{2} \) |
| 13 | \( 1 + 2.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.72iT - 17T^{2} \) |
| 19 | \( 1 + 6.76iT - 19T^{2} \) |
| 29 | \( 1 + 7.05iT - 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 - 7.38iT - 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 + 5.53T + 47T^{2} \) |
| 53 | \( 1 - 3.77iT - 53T^{2} \) |
| 59 | \( 1 - 4.18iT - 59T^{2} \) |
| 61 | \( 1 + 7.57iT - 61T^{2} \) |
| 67 | \( 1 + 5.95T + 67T^{2} \) |
| 71 | \( 1 - 9.66iT - 71T^{2} \) |
| 73 | \( 1 - 9.28iT - 73T^{2} \) |
| 79 | \( 1 - 12.3iT - 79T^{2} \) |
| 83 | \( 1 + 16.6iT - 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 9.94T + 97T^{2} \) |
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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
−10.64469977640112010582541394668, −9.965698113161667447816061575139, −9.403378313835177542170241096415, −8.454697104416725288959021485685, −7.59814019456121880920258588710, −6.81588925904195311131121310403, −5.88259729447415668657768773368, −4.08887569699897144719297783125, −3.27223153554126086239576233782, −2.50152715280366409456249619638,
0.096620942235420523086107798207, 1.75402850931228899980369620800, 3.23653051704641168764097842829, 3.92055646363299282028082218476, 5.72730899114223055181644023992, 6.78734252949222424989229025436, 7.39782081332816376624577180408, 8.342659619269881800280219874253, 9.223122940938321293934130522612, 9.511724756683322433361886026210