| L(s) = 1 | + 2-s + 3-s + 4-s − 0.709·5-s + 6-s + 0.335·7-s + 8-s + 9-s − 0.709·10-s − 1.82·11-s + 12-s − 1.79·13-s + 0.335·14-s − 0.709·15-s + 16-s − 1.87·17-s + 18-s − 19-s − 0.709·20-s + 0.335·21-s − 1.82·22-s + 2.28·23-s + 24-s − 4.49·25-s − 1.79·26-s + 27-s + 0.335·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.317·5-s + 0.408·6-s + 0.126·7-s + 0.353·8-s + 0.333·9-s − 0.224·10-s − 0.551·11-s + 0.288·12-s − 0.498·13-s + 0.0896·14-s − 0.183·15-s + 0.250·16-s − 0.453·17-s + 0.235·18-s − 0.229·19-s − 0.158·20-s + 0.0731·21-s − 0.389·22-s + 0.477·23-s + 0.204·24-s − 0.899·25-s − 0.352·26-s + 0.192·27-s + 0.0633·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.709T + 5T^{2} \) |
| 7 | \( 1 - 0.335T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 + 5.40T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 + 0.853T + 73T^{2} \) |
| 79 | \( 1 - 6.28T + 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66246493465762451790394951588, −7.07106877148582691566403026685, −6.34431808534212855821563133164, −5.29905242559131859200095608563, −4.92353076328117473521201527736, −3.85196016134003305032143923514, −3.43628738160358189968750448962, −2.37123258251995189365212460755, −1.73960403909687261905197458563, 0,
1.73960403909687261905197458563, 2.37123258251995189365212460755, 3.43628738160358189968750448962, 3.85196016134003305032143923514, 4.92353076328117473521201527736, 5.29905242559131859200095608563, 6.34431808534212855821563133164, 7.07106877148582691566403026685, 7.66246493465762451790394951588
