| L(s) = 1 | − 3-s + 3.11·5-s + 1.67·7-s + 9-s − 2.32·11-s + 4.78·13-s − 3.11·15-s + 0.327·17-s − 19-s − 1.67·21-s + 3.43·23-s + 4.67·25-s − 27-s + 1.34·29-s + 6.78·31-s + 2.32·33-s + 5.20·35-s + 6.09·37-s − 4.78·39-s − 4.22·41-s − 7.89·43-s + 3.11·45-s + 0.455·47-s − 4.20·49-s − 0.327·51-s − 2·53-s − 7.23·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.39·5-s + 0.632·7-s + 0.333·9-s − 0.701·11-s + 1.32·13-s − 0.803·15-s + 0.0793·17-s − 0.229·19-s − 0.365·21-s + 0.716·23-s + 0.934·25-s − 0.192·27-s + 0.249·29-s + 1.21·31-s + 0.405·33-s + 0.879·35-s + 1.00·37-s − 0.765·39-s − 0.659·41-s − 1.20·43-s + 0.463·45-s + 0.0664·47-s − 0.600·49-s − 0.0458·51-s − 0.274·53-s − 0.975·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118506159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118506159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.11T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 0.327T + 17T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 - 0.455T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 + 6.22T + 67T^{2} \) |
| 71 | \( 1 + 4.65T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.339573986104682908856118604501, −8.500203659433250273366234926373, −7.76191578006204514907329530523, −6.50336639345694938815739261908, −6.14660939278213983305687887759, −5.22796619576997248512163713129, −4.64092779262363786560694933433, −3.23447760045817588537431718689, −2.05945839533508440901102652840, −1.11422448131117460870432499412,
1.11422448131117460870432499412, 2.05945839533508440901102652840, 3.23447760045817588537431718689, 4.64092779262363786560694933433, 5.22796619576997248512163713129, 6.14660939278213983305687887759, 6.50336639345694938815739261908, 7.76191578006204514907329530523, 8.500203659433250273366234926373, 9.339573986104682908856118604501
