| L(s) = 1 | − 5.39·3-s − 7.70·5-s + 19.1·7-s + 2.11·9-s − 19.1·11-s + 43.0·13-s + 41.5·15-s − 114.·17-s + 88.0·19-s − 103.·21-s + 105.·23-s − 65.6·25-s + 134.·27-s + 223.·29-s + 174.·31-s + 103.·33-s − 147.·35-s + 37·37-s − 232.·39-s − 454.·41-s − 59.6·43-s − 16.3·45-s − 397.·47-s + 22.8·49-s + 617.·51-s − 161.·53-s + 147.·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s − 0.689·5-s + 1.03·7-s + 0.0784·9-s − 0.524·11-s + 0.918·13-s + 0.715·15-s − 1.63·17-s + 1.06·19-s − 1.07·21-s + 0.952·23-s − 0.525·25-s + 0.956·27-s + 1.43·29-s + 1.01·31-s + 0.545·33-s − 0.711·35-s + 0.164·37-s − 0.954·39-s − 1.73·41-s − 0.211·43-s − 0.0540·45-s − 1.23·47-s + 0.0666·49-s + 1.69·51-s − 0.418·53-s + 0.361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 + 5.39T + 27T^{2} \) |
| 5 | \( 1 + 7.70T + 125T^{2} \) |
| 7 | \( 1 - 19.1T + 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 174.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 454.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 59.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 155.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 356.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 546.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 624.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 905.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−10.15123518048182581613919060757, −8.682081099328437394423019377249, −8.194162662880371754598225687427, −7.04211758833951805124261334384, −6.17248156330236149663082234729, −5.04691956503276927602251351197, −4.50571573624483741262484679019, −3.01035398908230056613878886689, −1.31701585337783404563800010527, 0,
1.31701585337783404563800010527, 3.01035398908230056613878886689, 4.50571573624483741262484679019, 5.04691956503276927602251351197, 6.17248156330236149663082234729, 7.04211758833951805124261334384, 8.194162662880371754598225687427, 8.682081099328437394423019377249, 10.15123518048182581613919060757
