| L(s) = 1 | − i·2-s + 4-s − i·5-s + 4i·7-s − 3i·8-s − 10-s + (3 − 2i)13-s + 4·14-s − 16-s + 4·17-s − 4i·19-s − i·20-s + 8·23-s − 25-s + (−2 − 3i)26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 0.447i·5-s + 1.51i·7-s − 1.06i·8-s − 0.316·10-s + (0.832 − 0.554i)13-s + 1.06·14-s − 0.250·16-s + 0.970·17-s − 0.917i·19-s − 0.223i·20-s + 1.66·23-s − 0.200·25-s + (−0.392 − 0.588i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61335 - 0.863441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61335 - 0.863441i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−10.88312676434030412451832552717, −9.615783747013399980502574373000, −9.030520222824034382846136043118, −8.082261738308556363209029688189, −6.91793500011716014962991875231, −5.85510610391463206076543756997, −5.09232656613358683428711755582, −3.45877225236900606952240305079, −2.62747887660960708521776682054, −1.29285548526785188345452485075,
1.46890404849524784973073518000, 3.19517757967965801003814759240, 4.20093919910567923737747497790, 5.62286415620721175798825511894, 6.42676636859940756577644866212, 7.46974526252943020572585255289, 7.61373108658908892546154806376, 8.988771257619541843319366198001, 10.09316523462472833447834023152, 11.00447815983888266243270517554
