| L(s) = 1 | + 2.17·2-s − 3-s + 2.70·4-s + 3.70·5-s − 2.17·6-s − 7-s + 1.53·8-s + 9-s + 8.04·10-s − 1.07·11-s − 2.70·12-s + 0.921·13-s − 2.17·14-s − 3.70·15-s − 2.07·16-s + 0.290·17-s + 2.17·18-s + 19-s + 10.0·20-s + 21-s − 2.34·22-s − 7.60·23-s − 1.53·24-s + 8.75·25-s + 2·26-s − 27-s − 2.70·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.35·4-s + 1.65·5-s − 0.885·6-s − 0.377·7-s + 0.544·8-s + 0.333·9-s + 2.54·10-s − 0.325·11-s − 0.782·12-s + 0.255·13-s − 0.579·14-s − 0.957·15-s − 0.519·16-s + 0.0705·17-s + 0.511·18-s + 0.229·19-s + 2.24·20-s + 0.218·21-s − 0.498·22-s − 1.58·23-s − 0.314·24-s + 1.75·25-s + 0.392·26-s − 0.192·27-s − 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.001356903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001356903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 - 0.290T + 17T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 1.26T + 67T^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
| 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
−11.65547038783010857705466202572, −10.30318420308645919748399387561, −9.939662594440579997386953595328, −8.560184680708139936944214663155, −6.80201813165663562901847856350, −6.19288926420438796252546173097, −5.49323216642143360717978901928, −4.65826845090344499188971492955, −3.21591763127668172356756531832, −1.96711390167257975936627033293,
1.96711390167257975936627033293, 3.21591763127668172356756531832, 4.65826845090344499188971492955, 5.49323216642143360717978901928, 6.19288926420438796252546173097, 6.80201813165663562901847856350, 8.560184680708139936944214663155, 9.939662594440579997386953595328, 10.30318420308645919748399387561, 11.65547038783010857705466202572
