| L(s) = 1 | + 5.86·2-s + 26.2·3-s + 2.39·4-s + 25·5-s + 153.·6-s − 49·7-s − 173.·8-s + 444.·9-s + 146.·10-s − 676.·11-s + 62.7·12-s + 571.·13-s − 287.·14-s + 655.·15-s − 1.09e3·16-s + 1.90e3·17-s + 2.60e3·18-s − 1.36e3·19-s + 59.8·20-s − 1.28e3·21-s − 3.96e3·22-s − 1.74e3·23-s − 4.55e3·24-s + 625·25-s + 3.35e3·26-s + 5.29e3·27-s − 117.·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 1.68·3-s + 0.0747·4-s + 0.447·5-s + 1.74·6-s − 0.377·7-s − 0.959·8-s + 1.83·9-s + 0.463·10-s − 1.68·11-s + 0.125·12-s + 0.938·13-s − 0.391·14-s + 0.752·15-s − 1.06·16-s + 1.59·17-s + 1.89·18-s − 0.868·19-s + 0.0334·20-s − 0.635·21-s − 1.74·22-s − 0.688·23-s − 1.61·24-s + 0.200·25-s + 0.972·26-s + 1.39·27-s − 0.0282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.617672901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617672901\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 2 | \( 1 - 5.86T + 32T^{2} \) |
| 3 | \( 1 - 26.2T + 243T^{2} \) |
| 11 | \( 1 + 676.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 571.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.90e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 273.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 783.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.98e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.83e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.28e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.48e4T + 8.58e9T^{2} \) |
| 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
−15.05899113137250112004127168860, −14.17074420288463475920554163782, −13.30343888542224306683801773082, −12.68287702930683083437995777057, −10.28234102838813867572420366621, −9.016192469174879811567144102472, −7.81002597481251991654295122540, −5.68731164067951107936612277974, −3.79362765002237015652885181893, −2.59010437378028585205947659943,
2.59010437378028585205947659943, 3.79362765002237015652885181893, 5.68731164067951107936612277974, 7.81002597481251991654295122540, 9.016192469174879811567144102472, 10.28234102838813867572420366621, 12.68287702930683083437995777057, 13.30343888542224306683801773082, 14.17074420288463475920554163782, 15.05899113137250112004127168860
