| L(s) = 1 | − 2.71·3-s − 2.26·5-s + 1.04·7-s + 4.37·9-s + 4.92·11-s − 0.968·13-s + 6.14·15-s + 6.40·17-s − 6.87·19-s − 2.83·21-s − 5.06·23-s + 0.116·25-s − 3.72·27-s − 6.88·29-s + 5.16·31-s − 13.3·33-s − 2.36·35-s + 0.595·37-s + 2.63·39-s − 0.634·41-s + 2.58·43-s − 9.88·45-s + 3.05·47-s − 5.90·49-s − 17.3·51-s + 9.87·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 1.01·5-s + 0.394·7-s + 1.45·9-s + 1.48·11-s − 0.268·13-s + 1.58·15-s + 1.55·17-s − 1.57·19-s − 0.619·21-s − 1.05·23-s + 0.0232·25-s − 0.716·27-s − 1.27·29-s + 0.928·31-s − 2.32·33-s − 0.399·35-s + 0.0978·37-s + 0.421·39-s − 0.0990·41-s + 0.394·43-s − 1.47·45-s + 0.445·47-s − 0.843·49-s − 2.43·51-s + 1.35·53-s − 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 + 0.968T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 - 0.595T + 37T^{2} \) |
| 41 | \( 1 + 0.634T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 - 8.27T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 3.09T + 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| 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
−7.35968987820913449132238115334, −6.73875332730183050540166256337, −5.97094371565059385111432089305, −5.61497674100735122118214793918, −4.46044353992093473040417089439, −4.23419426247174466054285651557, −3.42217410855638558895868028001, −1.91480481925987373691272463107, −0.987571860926317605463907714326, 0,
0.987571860926317605463907714326, 1.91480481925987373691272463107, 3.42217410855638558895868028001, 4.23419426247174466054285651557, 4.46044353992093473040417089439, 5.61497674100735122118214793918, 5.97094371565059385111432089305, 6.73875332730183050540166256337, 7.35968987820913449132238115334
