| L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.70 − 0.292i)3-s − 1.00i·4-s + (−0.999 + 1.41i)6-s + (1 + i)7-s + (0.707 + 0.707i)8-s + (2.82 − i)9-s − 1.41i·11-s + (−0.292 − 1.70i)12-s − 1.41·14-s − 1.00·16-s + (−1.41 + 1.41i)17-s + (−1.29 + 2.70i)18-s + 4i·19-s + (2 + 1.41i)21-s + (1.00 + 1.00i)22-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.985 − 0.169i)3-s − 0.500i·4-s + (−0.408 + 0.577i)6-s + (0.377 + 0.377i)7-s + (0.250 + 0.250i)8-s + (0.942 − 0.333i)9-s − 0.426i·11-s + (−0.0845 − 0.492i)12-s − 0.377·14-s − 0.250·16-s + (−0.342 + 0.342i)17-s + (−0.304 + 0.638i)18-s + 0.917i·19-s + (0.436 + 0.308i)21-s + (0.213 + 0.213i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17655 + 0.239567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17655 + 0.239567i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (6 + 6i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−13.28266673646418027210768405969, −12.19777107817252047238037671000, −10.83679562939905047379668626769, −9.746015786771890271875642877818, −8.716489224863330776250912338929, −8.061669573307020052751037092460, −6.93392825734304078433315994792, −5.59018768909418191554155435772, −3.85518667600551828575971006169, −1.99628045672654958149132205408,
1.95237266924233552958915165429, 3.49215922874212530078715879765, 4.79237144463857601788691152308, 6.99471582093287865796436715133, 7.88489972593363744207462555435, 8.957241834901620649346050735535, 9.776710327613286970069863001227, 10.77599725997715439775079237345, 11.83750873494649873029244665804, 13.10652534396837725265735951781
