| L(s) = 1 | − 2.74·2-s − 4.19·3-s − 0.465·4-s − 5.57·5-s + 11.5·6-s + 6.88·7-s + 23.2·8-s − 9.42·9-s + 15.2·10-s + 11·11-s + 1.95·12-s − 18.8·14-s + 23.3·15-s − 60.0·16-s − 96.9·17-s + 25.8·18-s + 91.3·19-s + 2.59·20-s − 28.8·21-s − 30.1·22-s + 4.92·23-s − 97.4·24-s − 93.9·25-s + 152.·27-s − 3.20·28-s + 120.·29-s − 64.1·30-s + ⋯ |
| L(s) = 1 | − 0.970·2-s − 0.806·3-s − 0.0582·4-s − 0.498·5-s + 0.782·6-s + 0.371·7-s + 1.02·8-s − 0.349·9-s + 0.483·10-s + 0.301·11-s + 0.0469·12-s − 0.360·14-s + 0.402·15-s − 0.938·16-s − 1.38·17-s + 0.338·18-s + 1.10·19-s + 0.0290·20-s − 0.299·21-s − 0.292·22-s + 0.0446·23-s − 0.828·24-s − 0.751·25-s + 1.08·27-s − 0.0216·28-s + 0.772·29-s − 0.390·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.74T + 8T^{2} \) |
| 3 | \( 1 + 4.19T + 27T^{2} \) |
| 5 | \( 1 + 5.57T + 125T^{2} \) |
| 7 | \( 1 - 6.88T + 343T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.92T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 59.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 658.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 338.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 664.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 237.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 150.T + 9.12e5T^{2} \) |
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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
−8.538054024461767832209315942130, −7.86474654454951862658538226957, −7.04050270564171905429251648649, −6.23462249735609713781201357742, −5.12882706857636602772684007635, −4.57806989620103394152996202358, −3.47900033773648755747233557203, −2.01662560549749800013822406428, −0.851865316811273332271844374405, 0,
0.851865316811273332271844374405, 2.01662560549749800013822406428, 3.47900033773648755747233557203, 4.57806989620103394152996202358, 5.12882706857636602772684007635, 6.23462249735609713781201357742, 7.04050270564171905429251648649, 7.86474654454951862658538226957, 8.538054024461767832209315942130
