| L(s) = 1 | + 1.73·3-s − 1.54·5-s − 7-s + 0.0161·9-s − 3.07·11-s + 5.13·13-s − 2.68·15-s − 17-s + 2.95·19-s − 1.73·21-s + 0.956·23-s − 2.61·25-s − 5.18·27-s + 4.83·29-s − 3.98·31-s − 5.34·33-s + 1.54·35-s − 5.56·37-s + 8.91·39-s − 3.34·41-s − 1.96·43-s − 0.0249·45-s − 4.46·47-s + 49-s − 1.73·51-s − 7.31·53-s + 4.74·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.690·5-s − 0.377·7-s + 0.00537·9-s − 0.927·11-s + 1.42·13-s − 0.692·15-s − 0.242·17-s + 0.678·19-s − 0.378·21-s + 0.199·23-s − 0.523·25-s − 0.997·27-s + 0.898·29-s − 0.716·31-s − 0.930·33-s + 0.260·35-s − 0.914·37-s + 1.42·39-s − 0.522·41-s − 0.299·43-s − 0.00371·45-s − 0.651·47-s + 0.142·49-s − 0.243·51-s − 1.00·53-s + 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 - 0.956T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 + 5.56T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + 0.494T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 0.329T + 67T^{2} \) |
| 71 | \( 1 - 2.42T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 9.59T + 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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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
−8.225468054257371505036017396686, −7.62396643672805457224033199517, −6.79974885146387441717157620924, −5.90206247919535763181784291233, −5.09101526931573892034660582876, −3.98014149130606497813579784150, −3.37358806395957640211002193197, −2.75590996893853433981509946979, −1.56140328590977581426550347772, 0,
1.56140328590977581426550347772, 2.75590996893853433981509946979, 3.37358806395957640211002193197, 3.98014149130606497813579784150, 5.09101526931573892034660582876, 5.90206247919535763181784291233, 6.79974885146387441717157620924, 7.62396643672805457224033199517, 8.225468054257371505036017396686
