| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5869725302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5869725302\) |
| \(L(1)\) |
\(\approx\) |
\(0.4903903070\) |
| \(L(1)\) |
\(\approx\) |
\(0.4903903070\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2011 | \( 1 \) |
| 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
−19.44167710532034174673713206960, −18.795199440490354601649902051234, −18.27412759337075320503104985715, −17.490840184183667435076921565564, −17.08966376820287751735699079595, −16.21920759150620512463129143550, −15.75251643168943629092111822102, −15.048096339780971782741167593233, −13.57760509429361157884363322358, −12.96453042401194124707524587451, −12.48131358584505874744725395229, −11.1967591835191728076773441134, −10.7334603597687525289307295638, −10.23049679982057810045994063912, −9.32601922242884721055551251865, −8.8188924322186848043221328198, −7.62830611616709875272880339443, −6.696374284341075413466670792813, −6.246244612512553351879993066, −5.66240680578143094399159035058, −4.58489182327237320470412077272, −3.2400073706179842695957863972, −2.32559914715180992040795361510, −1.42972080304135414620158102274, −0.390932760648706106866736217,
0.390932760648706106866736217, 1.42972080304135414620158102274, 2.32559914715180992040795361510, 3.2400073706179842695957863972, 4.58489182327237320470412077272, 5.66240680578143094399159035058, 6.246244612512553351879993066, 6.696374284341075413466670792813, 7.62830611616709875272880339443, 8.8188924322186848043221328198, 9.32601922242884721055551251865, 10.23049679982057810045994063912, 10.7334603597687525289307295638, 11.1967591835191728076773441134, 12.48131358584505874744725395229, 12.96453042401194124707524587451, 13.57760509429361157884363322358, 15.048096339780971782741167593233, 15.75251643168943629092111822102, 16.21920759150620512463129143550, 17.08966376820287751735699079595, 17.490840184183667435076921565564, 18.27412759337075320503104985715, 18.795199440490354601649902051234, 19.44167710532034174673713206960
