| L(s) = 1 | + 3-s − 3.70·5-s + 7-s + 9-s + 4.51·11-s + 13-s − 3.70·15-s − 4.95·17-s + 3.27·19-s + 21-s + 8.14·23-s + 8.76·25-s + 27-s − 1.27·29-s − 10.2·31-s + 4.51·33-s − 3.70·35-s − 4.95·37-s + 39-s − 0.435·41-s + 8.14·43-s − 3.70·45-s − 5.24·47-s + 49-s − 4.95·51-s − 1.74·53-s − 16.7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.65·5-s + 0.377·7-s + 0.333·9-s + 1.36·11-s + 0.277·13-s − 0.957·15-s − 1.20·17-s + 0.751·19-s + 0.218·21-s + 1.69·23-s + 1.75·25-s + 0.192·27-s − 0.236·29-s − 1.83·31-s + 0.786·33-s − 0.627·35-s − 0.814·37-s + 0.160·39-s − 0.0680·41-s + 1.24·43-s − 0.552·45-s − 0.764·47-s + 0.142·49-s − 0.693·51-s − 0.239·53-s − 2.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911724791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911724791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 8.14T + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.95T + 37T^{2} \) |
| 41 | \( 1 + 0.435T + 41T^{2} \) |
| 43 | \( 1 - 8.14T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 1.74T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.67T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−8.402190306924003743007678511242, −7.58762891062732429239561262346, −7.12331927976461810334994236203, −6.47645448021669091630343636051, −5.14678292748948662956639622429, −4.44965224124263669392407818930, −3.68617090587645450551386620061, −3.27410385322269359602672137151, −1.90397859870249843010646188790, −0.77697084901374405030250800983,
0.77697084901374405030250800983, 1.90397859870249843010646188790, 3.27410385322269359602672137151, 3.68617090587645450551386620061, 4.44965224124263669392407818930, 5.14678292748948662956639622429, 6.47645448021669091630343636051, 7.12331927976461810334994236203, 7.58762891062732429239561262346, 8.402190306924003743007678511242
