| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6045112143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6045112143\) |
| \(L(1)\) |
\(\approx\) |
\(0.5993684564\) |
| \(L(1)\) |
\(\approx\) |
\(0.5993684564\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4643 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.36022691774310851115869218040, −17.2546633805338682003166672921, −16.65451154656901338486064126330, −15.93890144420522895511895760045, −15.45459154936457155375301339312, −14.94963954716908054306743047316, −14.24641044126571751010854771380, −13.06382668848002557094060070767, −12.55881664541429990566137757148, −12.10678558555104852419514107772, −10.86756340486381815159654295540, −10.512431699292164093626242058302, −9.549978982512850488112649461301, −9.302876346795195698994090848440, −8.20122762705611411941139153453, −7.95035119168315384764920306114, −7.166092150268481573303356057097, −6.73310178716362934550541513889, −5.57472797641244932037389154437, −4.54842941159274308915042095119, −3.5927382595576280341575734682, −2.90129521047993163051806993730, −2.52758596439009740550869087980, −1.34675217971808866490926731793, −0.30303888472367931511787100270,
0.30303888472367931511787100270, 1.34675217971808866490926731793, 2.52758596439009740550869087980, 2.90129521047993163051806993730, 3.5927382595576280341575734682, 4.54842941159274308915042095119, 5.57472797641244932037389154437, 6.73310178716362934550541513889, 7.166092150268481573303356057097, 7.95035119168315384764920306114, 8.20122762705611411941139153453, 9.302876346795195698994090848440, 9.549978982512850488112649461301, 10.512431699292164093626242058302, 10.86756340486381815159654295540, 12.10678558555104852419514107772, 12.55881664541429990566137757148, 13.06382668848002557094060070767, 14.24641044126571751010854771380, 14.94963954716908054306743047316, 15.45459154936457155375301339312, 15.93890144420522895511895760045, 16.65451154656901338486064126330, 17.2546633805338682003166672921, 18.36022691774310851115869218040
