| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 21-s + 2·22-s + 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8814254597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8814254597\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.57864766093536, −14.05737293650357, −13.31560340013000, −12.87365770756506, −12.19596971537553, −11.94704317444034, −11.26297588118378, −10.82324132124149, −10.53559464068275, −9.737375649825312, −9.442008187438180, −8.625391298386612, −8.208414139993967, −7.613831728298903, −7.205255540057514, −6.662536808795556, −5.902201764973311, −5.398308108987197, −4.834071623377958, −4.174119707541998, −3.416690573832806, −2.717732103981279, −1.981545374361188, −1.171333446847577, −0.4222663111323057,
0.4222663111323057, 1.171333446847577, 1.981545374361188, 2.717732103981279, 3.416690573832806, 4.174119707541998, 4.834071623377958, 5.398308108987197, 5.902201764973311, 6.662536808795556, 7.205255540057514, 7.613831728298903, 8.208414139993967, 8.625391298386612, 9.442008187438180, 9.737375649825312, 10.53559464068275, 10.82324132124149, 11.26297588118378, 11.94704317444034, 12.19596971537553, 12.87365770756506, 13.31560340013000, 14.05737293650357, 14.57864766093536
