L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.52 + 1.63i)5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (2.23 + 0.0743i)10-s − 2.14i·11-s + (2.16 − 2.16i)13-s − 1.00·14-s − 1.00·16-s + (4.46 − 4.46i)17-s + (1.63 − 1.52i)20-s + (−1.51 − 1.51i)22-s + (6.32 + 6.32i)23-s + (−0.332 + 4.98i)25-s − 3.05i·26-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.683 + 0.730i)5-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (0.706 + 0.0234i)10-s − 0.647i·11-s + (0.599 − 0.599i)13-s − 0.267·14-s − 0.250·16-s + (1.08 − 1.08i)17-s + (0.365 − 0.341i)20-s + (−0.323 − 0.323i)22-s + (1.31 + 1.31i)23-s + (−0.0664 + 0.997i)25-s − 0.599i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96655 - 0.926464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96655 - 0.926464i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.52 - 1.63i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 2.14iT - 11T^{2} \) |
| 13 | \( 1 + (-2.16 + 2.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.46 + 4.46i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-6.32 - 6.32i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + (5.13 + 5.13i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.56iT - 41T^{2} \) |
| 43 | \( 1 + (7.84 - 7.84i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.01 - 5.01i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.35 - 2.35i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.35T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + (3.84 + 3.84i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.420iT - 71T^{2} \) |
| 73 | \( 1 + (2.63 - 2.63i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.23iT - 79T^{2} \) |
| 83 | \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + (0.606 + 0.606i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61500998861415617079104326166, −9.793417127706419572912437479808, −9.028310182292378202215635591571, −7.68287508436169383860986145634, −6.74896884141069669881266659081, −5.80146093518682869897514758774, −5.06119274486268181807176787007, −3.40098260984895911252632608663, −2.96024461443265103933532551798, −1.23643976880131727076912142236,
1.58650892883983594001900882867, 3.10239416919458910532689596488, 4.43970966022175225717561703032, 5.20123423674023995272623245995, 6.24156692679608233049376092899, 6.85340694996879274054286897546, 8.289536796354017814407489988858, 8.738975896396418102785675448396, 9.824349165766300063089883741785, 10.56214593898237780765273673193