| L(s) = 1 | + (2.12 − 0.707i)5-s + (2 − 2i)7-s − 2.82i·11-s + (−1 − i)13-s + (2.82 + 2.82i)17-s + (2.82 − 2.82i)23-s + (3.99 − 3i)25-s − 4.24·29-s + 4·31-s + (2.82 − 5.65i)35-s + (−1 + i)37-s + 1.41i·41-s + (−8 − 8i)43-s + (5.65 + 5.65i)47-s − i·49-s + ⋯ |
| L(s) = 1 | + (0.948 − 0.316i)5-s + (0.755 − 0.755i)7-s − 0.852i·11-s + (−0.277 − 0.277i)13-s + (0.685 + 0.685i)17-s + (0.589 − 0.589i)23-s + (0.799 − 0.600i)25-s − 0.787·29-s + 0.718·31-s + (0.478 − 0.956i)35-s + (−0.164 + 0.164i)37-s + 0.220i·41-s + (−1.21 − 1.21i)43-s + (0.825 + 0.825i)47-s − 0.142i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432811921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432811921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
| good | 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (8 + 8i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−8.550229940858103598535546298077, −8.017873586701455155322697668499, −7.12185187821985961096770723154, −6.26518611169958586223094820605, −5.51281933142153884594811552347, −4.85985589064126355866025793999, −3.90084262446509947987247941471, −2.87967073465290727350828895351, −1.73060717974058337389289893991, −0.815106698019536773572083864024,
1.40917784723539574205453903669, 2.23237491302209856360856381507, 3.04348254917958838894078415680, 4.35424033944078833539137833988, 5.29135323062549076233384701762, 5.58610733837569787466745258135, 6.76658297597635693456002875043, 7.27328520699102392696489369395, 8.215850656026694483503678109518, 9.007978805773430893573194223578
