| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 11-s + 12-s − 2·13-s − 14-s + 3·15-s + 16-s + 3·17-s − 18-s + 5·19-s + 3·20-s + 21-s − 22-s + 23-s − 24-s + 4·25-s + 2·26-s + 27-s + 28-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 + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.14·19-s + 0.670·20-s + 0.218·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.817168007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.817168007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−10.09054883061974113404007026137, −9.343625063823747299058375739077, −8.992967941434406102315885951598, −7.69016020478956440151555647818, −7.17274969836490863356399952922, −5.88878294525968431722021150994, −5.20346724256150083623823335566, −3.56962216608174384840472434555, −2.34134556480991416306464799897, −1.42318167149940842052412714883,
1.42318167149940842052412714883, 2.34134556480991416306464799897, 3.56962216608174384840472434555, 5.20346724256150083623823335566, 5.88878294525968431722021150994, 7.17274969836490863356399952922, 7.69016020478956440151555647818, 8.992967941434406102315885951598, 9.343625063823747299058375739077, 10.09054883061974113404007026137
