| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 5·7-s − 8-s + 9-s + 3·10-s − 11-s − 12-s − 13-s − 5·14-s + 3·15-s + 16-s − 18-s − 3·20-s − 5·21-s + 22-s + 3·23-s + 24-s + 4·25-s + 26-s − 27-s + 5·28-s + 3·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.33·14-s + 0.774·15-s + 1/4·16-s − 0.235·18-s − 0.670·20-s − 1.09·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.944·28-s + 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834036899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834036899\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.36280955499691, −12.08628413951965, −11.55849143062057, −11.21711646708455, −11.04272473662476, −10.46867089705157, −10.16517659794323, −9.261178362832251, −8.826443186660740, −8.596314613658085, −7.740603342270645, −7.612509136318222, −7.467335540622298, −6.826158574647634, −5.925495586473846, −5.695436863896006, −4.934462792682851, −4.595590099116132, −4.163336233325506, −3.601567241758216, −2.771420887190381, −2.225071770418433, −1.610742266491470, −0.7983152577931185, −0.6004539923797294,
0.6004539923797294, 0.7983152577931185, 1.610742266491470, 2.225071770418433, 2.771420887190381, 3.601567241758216, 4.163336233325506, 4.595590099116132, 4.934462792682851, 5.695436863896006, 5.925495586473846, 6.826158574647634, 7.467335540622298, 7.612509136318222, 7.740603342270645, 8.596314613658085, 8.826443186660740, 9.261178362832251, 10.16517659794323, 10.46867089705157, 11.04272473662476, 11.21711646708455, 11.55849143062057, 12.08628413951965, 12.36280955499691
