| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s − 3·13-s + 3·14-s + 15-s + 16-s − 18-s + 7·19-s + 20-s − 3·21-s − 4·22-s − 4·23-s − 24-s + 25-s + 3·26-s + 27-s − 3·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.60·19-s + 0.223·20-s − 0.654·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.192·27-s − 0.566·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505217064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505217064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.524624994410450727187915509374, −9.249295909366009747505775581800, −8.047677215070149950467875831303, −7.31056974975953555539741575482, −6.49507952983038353412796529334, −5.78586207591239105101836731741, −4.34419939868507960414455974833, −3.25274694981497670131198956559, −2.41204094172111800841845912712, −1.01061161743011145946707204329,
1.01061161743011145946707204329, 2.41204094172111800841845912712, 3.25274694981497670131198956559, 4.34419939868507960414455974833, 5.78586207591239105101836731741, 6.49507952983038353412796529334, 7.31056974975953555539741575482, 8.047677215070149950467875831303, 9.249295909366009747505775581800, 9.524624994410450727187915509374
