| L(s) = 1 | + 3-s + 1.27·5-s + 0.298·7-s + 9-s − 2.92·11-s + 1.03·13-s + 1.27·15-s − 3.09·17-s + 3.33·19-s + 0.298·21-s − 5.64·23-s − 3.37·25-s + 27-s + 0.183·29-s − 10.8·31-s − 2.92·33-s + 0.381·35-s − 6.37·37-s + 1.03·39-s − 2.88·41-s + 2.09·43-s + 1.27·45-s + 6.91·47-s − 6.91·49-s − 3.09·51-s − 9.47·53-s − 3.73·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.570·5-s + 0.113·7-s + 0.333·9-s − 0.882·11-s + 0.287·13-s + 0.329·15-s − 0.750·17-s + 0.765·19-s + 0.0652·21-s − 1.17·23-s − 0.674·25-s + 0.192·27-s + 0.0340·29-s − 1.94·31-s − 0.509·33-s + 0.0644·35-s − 1.04·37-s + 0.166·39-s − 0.450·41-s + 0.319·43-s + 0.190·45-s + 1.00·47-s − 0.987·49-s − 0.433·51-s − 1.30·53-s − 0.503·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.298T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 5.64T + 23T^{2} \) |
| 29 | \( 1 - 0.183T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 - 2.81T + 97T^{2} \) |
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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
−7.69289331797934070605593856933, −7.21723869795015445368629951885, −6.22325476435578429493152286590, −5.60084400900963009606215585734, −4.88233244350136100458131251771, −3.93083144234863103450825210589, −3.20564528787995397505846048533, −2.21196658861218375725689753659, −1.64669172279654407006175398063, 0,
1.64669172279654407006175398063, 2.21196658861218375725689753659, 3.20564528787995397505846048533, 3.93083144234863103450825210589, 4.88233244350136100458131251771, 5.60084400900963009606215585734, 6.22325476435578429493152286590, 7.21723869795015445368629951885, 7.69289331797934070605593856933
