L(s) = 1 | + 0.414·2-s + 3-s − 1.82·4-s − 2.82·5-s + 0.414·6-s + 2.82·7-s − 1.58·8-s + 9-s − 1.17·10-s − 2·11-s − 1.82·12-s − 13-s + 1.17·14-s − 2.82·15-s + 3·16-s + 7.65·17-s + 0.414·18-s − 2.82·19-s + 5.17·20-s + 2.82·21-s − 0.828·22-s − 4·23-s − 1.58·24-s + 3.00·25-s − 0.414·26-s + 27-s − 5.17·28-s + ⋯ |
L(s) = 1 | + 0.292·2-s + 0.577·3-s − 0.914·4-s − 1.26·5-s + 0.169·6-s + 1.06·7-s − 0.560·8-s + 0.333·9-s − 0.370·10-s − 0.603·11-s − 0.527·12-s − 0.277·13-s + 0.313·14-s − 0.730·15-s + 0.750·16-s + 1.85·17-s + 0.0976·18-s − 0.648·19-s + 1.15·20-s + 0.617·21-s − 0.176·22-s − 0.834·23-s − 0.323·24-s + 0.600·25-s − 0.0812·26-s + 0.192·27-s − 0.977·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7964221304\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7964221304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−16.04950298307033131008500397708, −14.79138962396656964896705769639, −14.25172602711280113378671227511, −12.75072538387550213063268470792, −11.73521024606481280324460127322, −10.09067269620032594182897164525, −8.367796056521928312460730790959, −7.79055936858715051043363136838, −5.06473968936367901043856637263, −3.71045824363177121690250957726,
3.71045824363177121690250957726, 5.06473968936367901043856637263, 7.79055936858715051043363136838, 8.367796056521928312460730790959, 10.09067269620032594182897164525, 11.73521024606481280324460127322, 12.75072538387550213063268470792, 14.25172602711280113378671227511, 14.79138962396656964896705769639, 16.04950298307033131008500397708