| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 16.2·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 32.4·10-s + 34.2·11-s − 12·12-s − 49.1·13-s + 14·14-s − 48.6·15-s + 16·16-s + 8.58·17-s − 18·18-s − 5.80·19-s + 64.8·20-s + 21·21-s − 68.4·22-s + 23·23-s + 24·24-s + 138.·25-s + 98.2·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.45·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.02·10-s + 0.938·11-s − 0.288·12-s − 1.04·13-s + 0.267·14-s − 0.837·15-s + 0.250·16-s + 0.122·17-s − 0.235·18-s − 0.0701·19-s + 0.725·20-s + 0.218·21-s − 0.663·22-s + 0.208·23-s + 0.204·24-s + 1.10·25-s + 0.741·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 16.2T + 125T^{2} \) |
| 11 | \( 1 - 34.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.58T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.80T + 6.85e3T^{2} \) |
| 29 | \( 1 + 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 301.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 428.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 541.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 723.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 121.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 9.83T + 2.26e5T^{2} \) |
| 67 | \( 1 + 390.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 433.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 414.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 695.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 946.T + 9.12e5T^{2} \) |
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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
−9.355889846699772409995906270653, −8.731636264100528540804458908402, −7.22899698409686857688417174838, −6.79876377346737294226680495217, −5.77679890768281982047449990847, −5.22207368499007073526681208158, −3.69135281863740859036419038045, −2.26662373835993323808859328217, −1.45326890216498943402841816006, 0,
1.45326890216498943402841816006, 2.26662373835993323808859328217, 3.69135281863740859036419038045, 5.22207368499007073526681208158, 5.77679890768281982047449990847, 6.79876377346737294226680495217, 7.22899698409686857688417174838, 8.731636264100528540804458908402, 9.355889846699772409995906270653
