| L(s) = 1 | − 2-s − 3-s + 4-s − 2.56·5-s + 6-s − 0.973·7-s − 8-s + 9-s + 2.56·10-s + 3.22·11-s − 12-s + 5.35·13-s + 0.973·14-s + 2.56·15-s + 16-s − 3.23·17-s − 18-s + 19-s − 2.56·20-s + 0.973·21-s − 3.22·22-s − 5.36·23-s + 24-s + 1.58·25-s − 5.35·26-s − 27-s − 0.973·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s − 0.367·7-s − 0.353·8-s + 0.333·9-s + 0.811·10-s + 0.972·11-s − 0.288·12-s + 1.48·13-s + 0.260·14-s + 0.662·15-s + 0.250·16-s − 0.783·17-s − 0.235·18-s + 0.229·19-s − 0.573·20-s + 0.212·21-s − 0.687·22-s − 1.11·23-s + 0.204·24-s + 0.317·25-s − 1.05·26-s − 0.192·27-s − 0.183·28-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 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 0.973T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 + 5.36T + 23T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 59 | \( 1 + 0.929T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.61T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 0.264T + 97T^{2} \) |
| 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
−7.86198691260274962392642926965, −6.97646366025163330290594113909, −6.41421937585952722888590761374, −5.89844555265582756789373123428, −4.74003026037612403996722836203, −3.76875974148319710903588121923, −3.57483086908132417955989471952, −2.05263287008315425424756124801, −1.03300299019970064501218649114, 0,
1.03300299019970064501218649114, 2.05263287008315425424756124801, 3.57483086908132417955989471952, 3.76875974148319710903588121923, 4.74003026037612403996722836203, 5.89844555265582756789373123428, 6.41421937585952722888590761374, 6.97646366025163330290594113909, 7.86198691260274962392642926965
