L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.866 − 2.5i)7-s + (−0.499 − 0.866i)9-s + (−1.36 + 2.36i)11-s + 5.73·13-s + 0.999·15-s + (−3.36 + 5.83i)17-s + (−1.23 − 2.13i)19-s + (1.73 + 2i)21-s + (−0.633 − 1.09i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + 6.19·29-s + (3.23 − 5.59i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.327 − 0.944i)7-s + (−0.166 − 0.288i)9-s + (−0.411 + 0.713i)11-s + 1.58·13-s + 0.258·15-s + (−0.816 + 1.41i)17-s + (−0.282 − 0.489i)19-s + (0.377 + 0.436i)21-s + (−0.132 − 0.228i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + 1.15·29-s + (0.580 − 1.00i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.494718071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494718071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 11 | \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 + (3.36 - 5.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 + 2.13i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + (-3.23 + 5.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.19 + 7.26i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 8.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 2.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + (-2.33 + 4.03i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.69 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + (-4.56 - 7.90i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−9.294582866218538995942702846794, −8.432358219623757758266533586081, −7.932653501489679921970989883502, −6.76575841889937934428922247720, −6.13897844993060933111956763557, −5.02445757658769768998337860414, −4.18536572105183433001817553627, −3.76162379294957265087118159629, −2.10609265633179177477045624820, −0.75560813384656464123024486625,
1.03823794449243299381288736200, 2.42795099006746664575324357886, 3.23043317096804768331047393302, 4.51642273204914335442525004002, 5.50338409206048465541359940455, 6.18983717311986688547857813221, 6.87992626747246520371404251776, 7.937393040472896210701724706416, 8.569298056138184235674761007375, 9.120514314705957897948787763036