L(s) = 1 | − i·2-s − 1.73·3-s − 4-s + (−1.41 − 1.73i)5-s + 1.73i·6-s − 2.44i·7-s + i·8-s + 2.99·9-s + (−1.73 + 1.41i)10-s − 3.46i·11-s + 1.73·12-s − 2.44·14-s + (2.44 + 2.99i)15-s + 16-s − 7.07·17-s − 2.99i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.00·3-s − 0.5·4-s + (−0.632 − 0.774i)5-s + 0.707i·6-s − 0.925i·7-s + 0.353i·8-s + 0.999·9-s + (−0.547 + 0.447i)10-s − 1.04i·11-s + 0.500·12-s − 0.654·14-s + (0.632 + 0.774i)15-s + 0.250·16-s − 1.71·17-s − 0.707i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129344 + 0.206868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129344 + 0.206868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| 19 | \( 1 + (1 - 4.24i)T \) |
good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 9.79iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−10.57729012543963198011458605092, −9.354830637178846733704054569225, −8.526366768962406861172314104960, −7.46305064156373921928666945037, −6.42739823167623665293371053281, −5.22179598195474137391927274869, −4.38951164904599161191131814986, −3.54353303507515192840456397248, −1.42242707821544713951685268877, −0.16554038263611980153666059511,
2.36374246390627645213792062696, 4.12495435768697868066564933315, 4.93604343855221368635850804355, 5.98670179111842241755964823514, 6.97368579916703601005884244383, 7.26952172783058586828571489680, 8.722963977636558941143718664633, 9.459475100962939888148570578402, 10.69569935076034735008316528547, 11.21333006126991129220010066403