L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.204 − 0.204i)3-s + 1.00i·4-s + (−1.42 − 1.71i)5-s − 0.289i·6-s + (−0.707 + 0.707i)8-s − 2.91i·9-s + (0.204 − 2.22i)10-s + 5.62·11-s + (0.204 − 0.204i)12-s + (−1.42 − 1.42i)13-s + (−0.0593 + 0.645i)15-s − 1.00·16-s + (3.75 − 3.75i)17-s + (2.06 − 2.06i)18-s + 3.89·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.118 − 0.118i)3-s + 0.500i·4-s + (−0.639 − 0.768i)5-s − 0.118i·6-s + (−0.250 + 0.250i)8-s − 0.972i·9-s + (0.0647 − 0.704i)10-s + 1.69·11-s + (0.0591 − 0.0591i)12-s + (−0.396 − 0.396i)13-s + (−0.0153 + 0.166i)15-s − 0.250·16-s + (0.910 − 0.910i)17-s + (0.486 − 0.486i)18-s + 0.892·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62579 - 0.325545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62579 - 0.325545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.42 + 1.71i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.204 + 0.204i)T + 3iT^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + (1.42 + 1.42i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.75 + 3.75i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + (0.794 - 0.794i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.15iT - 29T^{2} \) |
| 31 | \( 1 + 3.84iT - 31T^{2} \) |
| 37 | \( 1 + (3.56 + 3.56i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.16 - 4.16i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.978 - 0.978i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 4.60iT - 61T^{2} \) |
| 67 | \( 1 + (-0.597 - 0.597i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.24iT - 79T^{2} \) |
| 83 | \( 1 + (-5.67 - 5.67i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + (6.63 - 6.63i)T - 97iT^{2} \) |
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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
−11.40166952456431935627777065568, −9.584400064819730628856028730774, −9.225430513000438587950694694867, −8.003405031671787871798077951387, −7.22492049188136751115102996980, −6.23891513932917386384107612771, −5.24610974381491038728894302443, −4.13756846431422879147871955410, −3.31605583529956669112137306465, −0.995629204024991212220061288654,
1.68933632908054256151371260313, 3.24391352131379105315704167814, 4.05830956953643844815848353309, 5.17803520422210065590873906949, 6.39442485721068598750632337097, 7.22461563533522279531240877613, 8.296902501141151134836452576810, 9.480872089176598134697731430135, 10.38115296675572096377736561517, 11.07732351249585290465039556568