| L(s) = 1 | − 0.895·3-s + 49·7-s − 242.·9-s + 384.·11-s − 940.·13-s + 1.39e3·17-s + 176.·19-s − 43.8·21-s − 1.64e3·23-s + 434.·27-s − 298.·29-s − 5.38e3·31-s − 344.·33-s + 1.15e3·37-s + 842.·39-s + 3.67e3·41-s + 1.43e4·43-s + 1.71e3·47-s + 2.40e3·49-s − 1.24e3·51-s + 4.74e3·53-s − 158.·57-s − 1.10e4·59-s + 4.54e4·61-s − 1.18e4·63-s − 3.48e4·67-s + 1.47e3·69-s + ⋯ |
| L(s) = 1 | − 0.0574·3-s + 0.377·7-s − 0.996·9-s + 0.958·11-s − 1.54·13-s + 1.16·17-s + 0.112·19-s − 0.0217·21-s − 0.647·23-s + 0.114·27-s − 0.0659·29-s − 1.00·31-s − 0.0550·33-s + 0.138·37-s + 0.0886·39-s + 0.341·41-s + 1.18·43-s + 0.113·47-s + 0.142·49-s − 0.0672·51-s + 0.232·53-s − 0.00645·57-s − 0.413·59-s + 1.56·61-s − 0.376·63-s − 0.949·67-s + 0.0372·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.788263684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788263684\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 0.895T + 243T^{2} \) |
| 11 | \( 1 - 384.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 940.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 176.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 298.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.67e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.43e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.74e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.21e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.81e4T + 8.58e9T^{2} \) |
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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
−9.618820334591988418648669369720, −8.909576001164224618865836052771, −7.87080690066052922123942388709, −7.19953564645141269566710714835, −5.98472197682324453143971721229, −5.27964547651636781321071338916, −4.18108608848066476350087551430, −3.05004483016721905022707032826, −1.96087567477663958810540104794, −0.62020792377689727028800173445,
0.62020792377689727028800173445, 1.96087567477663958810540104794, 3.05004483016721905022707032826, 4.18108608848066476350087551430, 5.27964547651636781321071338916, 5.98472197682324453143971721229, 7.19953564645141269566710714835, 7.87080690066052922123942388709, 8.909576001164224618865836052771, 9.618820334591988418648669369720
