| L(s) = 1 | − 1.59·2-s − 5.44·4-s + 17.2·5-s + 7·7-s + 21.5·8-s − 27.6·10-s − 11·11-s + 46.1·13-s − 11.1·14-s + 9.11·16-s − 19.8·17-s + 76.5·19-s − 93.9·20-s + 17.5·22-s + 163.·23-s + 173.·25-s − 73.7·26-s − 38.0·28-s − 158.·29-s + 170.·31-s − 186.·32-s + 31.7·34-s + 120.·35-s − 245.·37-s − 122.·38-s + 371.·40-s + 3.33·41-s + ⋯ |
| L(s) = 1 | − 0.565·2-s − 0.680·4-s + 1.54·5-s + 0.377·7-s + 0.950·8-s − 0.874·10-s − 0.301·11-s + 0.983·13-s − 0.213·14-s + 0.142·16-s − 0.283·17-s + 0.924·19-s − 1.05·20-s + 0.170·22-s + 1.47·23-s + 1.38·25-s − 0.556·26-s − 0.257·28-s − 1.01·29-s + 0.989·31-s − 1.03·32-s + 0.160·34-s + 0.584·35-s − 1.09·37-s − 0.522·38-s + 1.46·40-s + 0.0127·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.941850451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941850451\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.59T + 8T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 13 | \( 1 - 46.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.33T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 21.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 618.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 929.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 868.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 152.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 100.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−9.975134798553365431949019210853, −9.153512138391453150467368880748, −8.669914983972517088752935633477, −7.59528417470824914659072702850, −6.49827509700104427821851259377, −5.45156880061480114105859377419, −4.84988767304440927799871452842, −3.35576340975389550884811278914, −1.88441430002707892073965282103, −0.954094278141210649750337880694,
0.954094278141210649750337880694, 1.88441430002707892073965282103, 3.35576340975389550884811278914, 4.84988767304440927799871452842, 5.45156880061480114105859377419, 6.49827509700104427821851259377, 7.59528417470824914659072702850, 8.669914983972517088752935633477, 9.153512138391453150467368880748, 9.975134798553365431949019210853
