L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s + 4·11-s − 12-s + 4·13-s + 14-s + 16-s + 8·17-s + 2·18-s + 5·19-s + 21-s − 4·22-s + 24-s − 4·26-s + 5·27-s − 28-s + 7·29-s − 4·31-s − 32-s − 4·33-s − 8·34-s − 2·36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.471·18-s + 1.14·19-s + 0.218·21-s − 0.852·22-s + 0.204·24-s − 0.784·26-s + 0.962·27-s − 0.188·28-s + 1.29·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s − 1.37·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.303491132\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.303491132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.98801985316447, −12.47974551099846, −12.03715514789840, −11.63570393938029, −11.34481613906729, −10.79444148818663, −10.28447278097039, −9.735130372412655, −9.435056029447699, −8.902083858801301, −8.366553598024313, −7.928711078940349, −7.440452133003291, −6.748886388827287, −6.290832265686305, −6.033924542329194, −5.379571222834551, −5.010365690280780, −4.058006575997840, −3.397194572684238, −3.272249359334844, −2.444707945641417, −1.513426257201974, −0.9551043169427600, −0.6720160145512873,
0.6720160145512873, 0.9551043169427600, 1.513426257201974, 2.444707945641417, 3.272249359334844, 3.397194572684238, 4.058006575997840, 5.010365690280780, 5.379571222834551, 6.033924542329194, 6.290832265686305, 6.748886388827287, 7.440452133003291, 7.928711078940349, 8.366553598024313, 8.902083858801301, 9.435056029447699, 9.735130372412655, 10.28447278097039, 10.79444148818663, 11.34481613906729, 11.63570393938029, 12.03715514789840, 12.47974551099846, 12.98801985316447