| L(s) = 1 | + 3-s − 0.129·5-s + 2.39·7-s + 9-s + 2.75·11-s − 0.964·13-s − 0.129·15-s − 6.11·17-s + 6.08·19-s + 2.39·21-s − 5.83·23-s − 4.98·25-s + 27-s + 6.37·29-s + 7.82·31-s + 2.75·33-s − 0.309·35-s + 1.11·37-s − 0.964·39-s + 4.38·41-s + 7.94·43-s − 0.129·45-s + 5.12·47-s − 1.26·49-s − 6.11·51-s − 1.79·53-s − 0.357·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0578·5-s + 0.905·7-s + 0.333·9-s + 0.832·11-s − 0.267·13-s − 0.0334·15-s − 1.48·17-s + 1.39·19-s + 0.522·21-s − 1.21·23-s − 0.996·25-s + 0.192·27-s + 1.18·29-s + 1.40·31-s + 0.480·33-s − 0.0523·35-s + 0.183·37-s − 0.154·39-s + 0.685·41-s + 1.21·43-s − 0.0192·45-s + 0.747·47-s − 0.180·49-s − 0.855·51-s − 0.246·53-s − 0.0481·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.964360356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.964360356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 0.129T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 0.964T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 6.37T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
| 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
−7.915905821056493703729768262580, −7.69288817786573907044547165145, −6.61956492004962136766060733303, −6.11466567262712414017788929264, −4.98926365400376825136617436328, −4.40667236321206856101928321145, −3.75177035424861940356782760595, −2.65176475736434557547049248089, −1.94335721694005217050032417539, −0.914005085916133644744598845585,
0.914005085916133644744598845585, 1.94335721694005217050032417539, 2.65176475736434557547049248089, 3.75177035424861940356782760595, 4.40667236321206856101928321145, 4.98926365400376825136617436328, 6.11466567262712414017788929264, 6.61956492004962136766060733303, 7.69288817786573907044547165145, 7.915905821056493703729768262580
