| L(s) = 1 | + 21.3·3-s − 101.·5-s − 45.0·7-s + 210.·9-s − 391.·11-s + 955.·13-s − 2.16e3·15-s + 1.39e3·17-s + 184.·19-s − 959.·21-s − 812.·23-s + 7.15e3·25-s − 682.·27-s + 4.46e3·29-s − 2.19e3·31-s − 8.33e3·33-s + 4.56e3·35-s − 4.27e3·37-s + 2.03e4·39-s − 18.0·41-s − 8.04e3·43-s − 2.13e4·45-s + 1.78e4·47-s − 1.47e4·49-s + 2.96e4·51-s − 7.97e3·53-s + 3.96e4·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s − 1.81·5-s − 0.347·7-s + 0.868·9-s − 0.974·11-s + 1.56·13-s − 2.47·15-s + 1.16·17-s + 0.117·19-s − 0.474·21-s − 0.320·23-s + 2.28·25-s − 0.180·27-s + 0.985·29-s − 0.409·31-s − 1.33·33-s + 0.630·35-s − 0.513·37-s + 2.14·39-s − 0.00167·41-s − 0.663·43-s − 1.57·45-s + 1.17·47-s − 0.879·49-s + 1.59·51-s − 0.390·53-s + 1.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 - 21.3T + 243T^{2} \) |
| 5 | \( 1 + 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 45.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 391.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 955.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 184.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 812.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 18.0T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.04e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.97e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.01e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−8.492245861375178054705251136420, −8.031039752621322140472207466960, −7.58991308038008769530780832764, −6.47907591113563026568032556266, −5.13976453749821529778318570858, −3.89847381176497470740431596665, −3.47657529834809180546733767019, −2.75011826062746277892422109736, −1.20777433895735679220751497161, 0,
1.20777433895735679220751497161, 2.75011826062746277892422109736, 3.47657529834809180546733767019, 3.89847381176497470740431596665, 5.13976453749821529778318570858, 6.47907591113563026568032556266, 7.58991308038008769530780832764, 8.031039752621322140472207466960, 8.492245861375178054705251136420
