L(s) = 1 | + (0.5 + 0.866i)3-s + (−1.63 + 2.83i)5-s + (−1.5 + 2.17i)7-s + (−0.499 + 0.866i)9-s + (−1.63 − 2.83i)11-s + 6.27·13-s − 3.27·15-s + (2 + 3.46i)17-s + (3.13 − 5.43i)19-s + (−2.63 − 0.209i)21-s + (−2 + 3.46i)23-s + (−2.86 − 4.95i)25-s − 0.999·27-s + 5.27·29-s + (0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.732 + 1.26i)5-s + (−0.566 + 0.823i)7-s + (−0.166 + 0.288i)9-s + (−0.493 − 0.855i)11-s + 1.74·13-s − 0.845·15-s + (0.485 + 0.840i)17-s + (0.719 − 1.24i)19-s + (−0.575 − 0.0456i)21-s + (−0.417 + 0.722i)23-s + (−0.572 − 0.991i)25-s − 0.192·27-s + 0.979·29-s + (0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755810 + 0.743946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755810 + 0.743946i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.5 - 2.17i)T \) |
good | 5 | \( 1 + (1.63 - 2.83i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.63 + 2.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.13 + 5.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.13 + 1.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.63 + 8.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.137 - 0.238i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.77 - 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + (-5.27 + 9.13i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
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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
−13.22303832855252912954438821210, −11.74239549734248389398112833984, −11.04755870729834060697208629278, −10.21090441376872129078725427074, −8.876443902352536423344811958797, −8.049835981511005071143283555717, −6.65159635195609449237979103670, −5.63810465534148831552591221915, −3.66089444704539507203586502099, −2.97240022167920627026528298975,
1.09291297764788396020458372580, 3.49941947805212402969333979308, 4.66836700088757003073783697242, 6.21992685391274186500494114986, 7.59830591293046006325353882365, 8.248039147669915871251344272830, 9.394922777227781402507320912305, 10.50739952010498938262130732539, 11.90945748248805802556409302081, 12.55101789207877130792679053503