L(s) = 1 | − 1.76i·3-s + (0.432 + 2.19i)5-s + i·7-s − 0.103·9-s + 0.626·11-s + 5.49i·13-s + (3.86 − 0.761i)15-s − 0.896i·17-s + 6.38·19-s + 1.76·21-s + 3.72i·23-s + (−4.62 + 1.89i)25-s − 5.10i·27-s + 7.87·29-s − 7.52·31-s + ⋯ |
L(s) = 1 | − 1.01i·3-s + (0.193 + 0.981i)5-s + 0.377i·7-s − 0.0343·9-s + 0.188·11-s + 1.52i·13-s + (0.997 − 0.196i)15-s − 0.217i·17-s + 1.46·19-s + 0.384·21-s + 0.777i·23-s + (−0.925 + 0.379i)25-s − 0.982i·27-s + 1.46·29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56061 + 0.152300i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56061 + 0.152300i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.432 - 2.19i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 1.76iT - 3T^{2} \) |
| 11 | \( 1 - 0.626T + 11T^{2} \) |
| 13 | \( 1 - 5.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 + 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.87iT - 47T^{2} \) |
| 53 | \( 1 + 6.77iT - 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 + 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 5.67T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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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
−11.02108733671631045389214359724, −9.767452910354862075574358939766, −9.170435535228704090264152644551, −7.80570290190444910658883247571, −7.10991957381176636260174258407, −6.51453785623653309798139331614, −5.47866603137611567964675639092, −3.96315645690292442856014266834, −2.63661425687114358251869747208, −1.55334613828464765983705725699,
1.05132838118880546337476731993, 3.08378592155570602746587790482, 4.19771308081562084190290842048, 5.03825537725062467941510084184, 5.81092254594430027288587054585, 7.28365238674658218140613982933, 8.253873111715598320039824054160, 9.096815937711357390610416460699, 10.00512524905029972391007520060, 10.39575671753134258058834243470