L(s) = 1 | − 1.29·3-s + 0.367·5-s + 7-s − 1.31·9-s − 11-s + 13-s − 0.478·15-s − 3.16·17-s − 2.94·19-s − 1.29·21-s − 2.33·23-s − 4.86·25-s + 5.60·27-s − 5.14·29-s − 5.54·31-s + 1.29·33-s + 0.367·35-s + 2.80·37-s − 1.29·39-s + 4.11·41-s + 3.28·43-s − 0.482·45-s − 1.06·47-s + 49-s + 4.11·51-s + 1.85·53-s − 0.367·55-s + ⋯ |
L(s) = 1 | − 0.750·3-s + 0.164·5-s + 0.377·7-s − 0.436·9-s − 0.301·11-s + 0.277·13-s − 0.123·15-s − 0.767·17-s − 0.676·19-s − 0.283·21-s − 0.486·23-s − 0.972·25-s + 1.07·27-s − 0.955·29-s − 0.995·31-s + 0.226·33-s + 0.0621·35-s + 0.461·37-s − 0.208·39-s + 0.642·41-s + 0.500·43-s − 0.0718·45-s − 0.154·47-s + 0.142·49-s + 0.576·51-s + 0.255·53-s − 0.0495·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9332571107\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9332571107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 5 | \( 1 - 0.367T + 5T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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
−7.84876629114027218962331458579, −7.08504475412674901858762251238, −6.26126932867098865745882456190, −5.79268446530268403688549482687, −5.18530809814256070662240247916, −4.34936944416241611272868141770, −3.66311080566121857659951962808, −2.49560348373047932845861779396, −1.81914571530957046968042464747, −0.48035809099157947999530886665,
0.48035809099157947999530886665, 1.81914571530957046968042464747, 2.49560348373047932845861779396, 3.66311080566121857659951962808, 4.34936944416241611272868141770, 5.18530809814256070662240247916, 5.79268446530268403688549482687, 6.26126932867098865745882456190, 7.08504475412674901858762251238, 7.84876629114027218962331458579