| L(s) = 1 | − 0.0882·2-s − 2.89·3-s − 1.99·4-s − 5-s + 0.255·6-s − 1.36·7-s + 0.352·8-s + 5.36·9-s + 0.0882·10-s + 0.593·11-s + 5.76·12-s − 3.39·13-s + 0.120·14-s + 2.89·15-s + 3.95·16-s − 3.62·17-s − 0.473·18-s − 4.64·19-s + 1.99·20-s + 3.93·21-s − 0.0523·22-s − 1.01·24-s + 25-s + 0.299·26-s − 6.83·27-s + 2.71·28-s − 8.12·29-s + ⋯ |
| L(s) = 1 | − 0.0624·2-s − 1.66·3-s − 0.996·4-s − 0.447·5-s + 0.104·6-s − 0.514·7-s + 0.124·8-s + 1.78·9-s + 0.0279·10-s + 0.178·11-s + 1.66·12-s − 0.940·13-s + 0.0321·14-s + 0.746·15-s + 0.988·16-s − 0.880·17-s − 0.111·18-s − 1.06·19-s + 0.445·20-s + 0.858·21-s − 0.0111·22-s − 0.208·24-s + 0.200·25-s + 0.0587·26-s − 1.31·27-s + 0.512·28-s − 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01161147206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01161147206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.0882T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.593T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 + 3.47T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 79 | \( 1 + 4.09T + 79T^{2} \) |
| 83 | \( 1 + 5.63T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.895181035297379389330771513566, −8.168479514859973591806601546009, −6.93141434241627900093952826541, −6.72485324051013837297431499622, −5.51806880190327023246800640681, −5.08858507342080149960741043151, −4.28558279318602325486685670595, −3.53125281461191136831908428020, −1.76553349745504795325484308464, −0.07298873586950031317359912710,
0.07298873586950031317359912710, 1.76553349745504795325484308464, 3.53125281461191136831908428020, 4.28558279318602325486685670595, 5.08858507342080149960741043151, 5.51806880190327023246800640681, 6.72485324051013837297431499622, 6.93141434241627900093952826541, 8.168479514859973591806601546009, 8.895181035297379389330771513566
