| L(s) = 1 | − 1.45·2-s + 1.34·3-s + 0.104·4-s + 3.00·5-s − 1.95·6-s − 1.88·7-s + 2.74·8-s − 1.17·9-s − 4.35·10-s + 3.71·11-s + 0.141·12-s + 2.53·13-s + 2.73·14-s + 4.05·15-s − 4.19·16-s + 3.78·17-s + 1.71·18-s − 6.83·19-s + 0.313·20-s − 2.54·21-s − 5.39·22-s + 3.18·23-s + 3.71·24-s + 4.00·25-s − 3.67·26-s − 5.63·27-s − 0.197·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.779·3-s + 0.0523·4-s + 1.34·5-s − 0.799·6-s − 0.713·7-s + 0.972·8-s − 0.392·9-s − 1.37·10-s + 1.12·11-s + 0.0407·12-s + 0.703·13-s + 0.731·14-s + 1.04·15-s − 1.04·16-s + 0.918·17-s + 0.403·18-s − 1.56·19-s + 0.0702·20-s − 0.555·21-s − 1.15·22-s + 0.663·23-s + 0.757·24-s + 0.801·25-s − 0.721·26-s − 1.08·27-s − 0.0373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.406931887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406931887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 3.00T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 9.55T + 43T^{2} \) |
| 47 | \( 1 + 0.876T + 47T^{2} \) |
| 53 | \( 1 + 0.975T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 0.866T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 0.894T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
| 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
−9.734907426804094606611807388883, −9.158422330691964550246936019634, −8.642182567647459052243632748785, −7.86358706778464720635600617806, −6.48253993002661791346321778041, −6.13492914935288786869808999618, −4.66477475566821071218503625199, −3.43018874541326611615890584636, −2.28787806475086871882884943148, −1.13340273188350873488369810929,
1.13340273188350873488369810929, 2.28787806475086871882884943148, 3.43018874541326611615890584636, 4.66477475566821071218503625199, 6.13492914935288786869808999618, 6.48253993002661791346321778041, 7.86358706778464720635600617806, 8.642182567647459052243632748785, 9.158422330691964550246936019634, 9.734907426804094606611807388883
