| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 12-s + 3·13-s + 14-s − 3·15-s + 16-s + 5·17-s − 18-s + 8·19-s − 3·20-s − 21-s + 23-s − 24-s + 4·25-s − 3·26-s + 27-s − 28-s + 9·29-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 − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s + 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.83·19-s − 0.670·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342243916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342243916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.64801218767339, −13.12004670471069, −12.36336472159602, −12.08624250619928, −11.73539773825556, −11.16456942650363, −10.60564938620981, −10.07214553999615, −9.757675732316582, −8.946355841331327, −8.721344551121003, −8.193882449538988, −7.633312224461418, −7.305071083219730, −6.962819337720982, −6.092907243964019, −5.625360986503885, −4.861990074295121, −4.281310668384777, −3.403366579116042, −3.282589308618919, −2.905899190409854, −1.611275089900829, −1.265922871317238, −0.4161122825672521,
0.4161122825672521, 1.265922871317238, 1.611275089900829, 2.905899190409854, 3.282589308618919, 3.403366579116042, 4.281310668384777, 4.861990074295121, 5.625360986503885, 6.092907243964019, 6.962819337720982, 7.305071083219730, 7.633312224461418, 8.193882449538988, 8.721344551121003, 8.946355841331327, 9.757675732316582, 10.07214553999615, 10.60564938620981, 11.16456942650363, 11.73539773825556, 12.08624250619928, 12.36336472159602, 13.12004670471069, 13.64801218767339
