L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (2.63 − 0.189i)7-s + 0.999i·8-s + (−0.866 − 0.499i)10-s + (1.32 + 0.766i)11-s − 1.48i·13-s + (−2.19 + 1.48i)14-s + (−0.5 − 0.866i)16-s + (−1.21 + 2.10i)17-s + (4.21 − 2.43i)19-s + 0.999·20-s − 1.53·22-s + (−0.232 + 0.133i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (0.997 − 0.0716i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + (0.400 + 0.230i)11-s − 0.411i·13-s + (−0.585 + 0.396i)14-s + (−0.125 − 0.216i)16-s + (−0.294 + 0.510i)17-s + (0.966 − 0.557i)19-s + 0.223·20-s − 0.326·22-s + (−0.0483 + 0.0279i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25739 + 0.409819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25739 + 0.409819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.63 + 0.189i)T \) |
good | 11 | \( 1 + (-1.32 - 0.766i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 17 | \( 1 + (1.21 - 2.10i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 2.43i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.232 - 0.133i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (0.717 + 0.414i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 4.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + (-3.72 - 6.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.00 + 1.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.12 - 5.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.73 + 3.31i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.01 - 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (-0.297 - 0.171i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.22 - 9.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 + (7.98 + 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−10.68855516146593972119507932781, −9.725328264552977655420206239271, −8.953130789869264796197409405914, −7.958018717342740309908185522785, −7.33550314101255455047370001398, −6.29242851343855283771835998910, −5.35082285423107949979916742368, −4.24622822290652179464359973245, −2.65481048977226999237359752527, −1.29142031153725480086569818891,
1.13735285211416314601570748270, 2.32688816473825851316964627701, 3.83101880751670399565876196283, 4.92736665609920557162665560870, 5.97607254159231184305910715761, 7.24111499407396135928333530436, 7.979407896987565257795442934936, 8.945220607039721534204913865827, 9.463565466594795544779540231963, 10.55017785497320312491730521749