| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 5·11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 7·17-s − 18-s + 19-s − 20-s − 21-s + 5·22-s − 8·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.06·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.08315115758319, −14.22347906005181, −13.61284051485322, −13.10864231933670, −12.70641086961533, −11.91821250739802, −11.49476913784995, −11.30649758864409, −10.42769222657814, −10.20572521296076, −9.797660238866074, −8.700956854422093, −8.613208703089313, −7.888402022773084, −7.440174360900149, −6.916078357568377, −6.316241971466016, −5.586209141371210, −5.171836043137242, −4.430302580190587, −3.937676982322125, −2.979915937369983, −2.228318985740913, −1.839134590400060, −0.6002022432301194, 0,
0.6002022432301194, 1.839134590400060, 2.228318985740913, 2.979915937369983, 3.937676982322125, 4.430302580190587, 5.171836043137242, 5.586209141371210, 6.316241971466016, 6.916078357568377, 7.440174360900149, 7.888402022773084, 8.613208703089313, 8.700956854422093, 9.797660238866074, 10.20572521296076, 10.42769222657814, 11.30649758864409, 11.49476913784995, 11.91821250739802, 12.70641086961533, 13.10864231933670, 13.61284051485322, 14.22347906005181, 15.08315115758319
