| L(s) = 1 | − 0.211·3-s + 4.34·5-s + 3.73·7-s − 2.95·9-s + 4.64·11-s − 2.21·13-s − 0.919·15-s + 2.91·17-s − 4.29·19-s − 0.790·21-s + 6.56·23-s + 13.8·25-s + 1.26·27-s − 7.98·29-s + 31-s − 0.983·33-s + 16.2·35-s − 2.55·37-s + 0.468·39-s − 4.90·41-s + 11.6·43-s − 12.8·45-s − 5.15·47-s + 6.91·49-s − 0.618·51-s − 4.96·53-s + 20.1·55-s + ⋯ |
| L(s) = 1 | − 0.122·3-s + 1.94·5-s + 1.41·7-s − 0.985·9-s + 1.40·11-s − 0.613·13-s − 0.237·15-s + 0.708·17-s − 0.984·19-s − 0.172·21-s + 1.36·23-s + 2.77·25-s + 0.242·27-s − 1.48·29-s + 0.179·31-s − 0.171·33-s + 2.73·35-s − 0.419·37-s + 0.0750·39-s − 0.765·41-s + 1.77·43-s − 1.91·45-s − 0.751·47-s + 0.988·49-s − 0.0865·51-s − 0.681·53-s + 2.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.874608851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.874608851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 0.211T + 3T^{2} \) |
| 5 | \( 1 - 4.34T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 + 4.96T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 0.774T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 9.34T + 89T^{2} \) |
| 97 | \( 1 - 8.45T + 97T^{2} \) |
| 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
−9.106882737556802793680769998301, −8.702321132004516942995400873698, −7.58683926092320843434494733889, −6.60454115112184752120578587707, −5.90662535633196893158110580937, −5.26225261598326274843838874956, −4.55092389042069197892046670710, −3.06301847583068097050076055182, −1.99625223973221999234414115698, −1.33704074189966684221141952202,
1.33704074189966684221141952202, 1.99625223973221999234414115698, 3.06301847583068097050076055182, 4.55092389042069197892046670710, 5.26225261598326274843838874956, 5.90662535633196893158110580937, 6.60454115112184752120578587707, 7.58683926092320843434494733889, 8.702321132004516942995400873698, 9.106882737556802793680769998301
