| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 7-s + 9-s + 5·11-s − 2·12-s + 13-s − 2·14-s − 4·16-s + 7·17-s − 2·18-s − 6·19-s − 21-s − 10·22-s − 3·23-s − 2·26-s − 27-s + 2·28-s + 2·29-s + 2·31-s + 8·32-s − 5·33-s − 14·34-s + 2·36-s − 7·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s + 0.277·13-s − 0.534·14-s − 16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s − 0.218·21-s − 2.13·22-s − 0.625·23-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.359·31-s + 1.41·32-s − 0.870·33-s − 2.40·34-s + 1/3·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7222752146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7222752146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.916578891756839924454082102240, −9.251312219508712135294968453202, −8.378803253964416403880030335796, −7.74165901414068773623906798462, −6.71123534518408333297550205349, −6.06112740203192888872476914574, −4.74297496906004716382997842488, −3.72637631783812950059768618289, −1.88115677514491653875272833927, −0.902342689790264195025876590163,
0.902342689790264195025876590163, 1.88115677514491653875272833927, 3.72637631783812950059768618289, 4.74297496906004716382997842488, 6.06112740203192888872476914574, 6.71123534518408333297550205349, 7.74165901414068773623906798462, 8.378803253964416403880030335796, 9.251312219508712135294968453202, 9.916578891756839924454082102240
