L(s) = 1 | + 1.21·2-s + 2.90·3-s − 0.525·4-s + 3.52·6-s − 1.52·7-s − 3.06·8-s + 5.42·9-s + 4.90·11-s − 1.52·12-s + 6.42·13-s − 1.85·14-s − 2.67·16-s − 2.14·17-s + 6.59·18-s + 2.28·19-s − 4.42·21-s + 5.95·22-s − 6.90·23-s − 8.90·24-s + 7.80·26-s + 7.05·27-s + 0.801·28-s + 29-s + 1.71·31-s + 2.88·32-s + 14.2·33-s − 2.60·34-s + ⋯ |
L(s) = 1 | + 0.858·2-s + 1.67·3-s − 0.262·4-s + 1.43·6-s − 0.576·7-s − 1.08·8-s + 1.80·9-s + 1.47·11-s − 0.440·12-s + 1.78·13-s − 0.495·14-s − 0.668·16-s − 0.520·17-s + 1.55·18-s + 0.523·19-s − 0.966·21-s + 1.26·22-s − 1.43·23-s − 1.81·24-s + 1.53·26-s + 1.35·27-s + 0.151·28-s + 0.185·29-s + 0.308·31-s + 0.510·32-s + 2.47·33-s − 0.447·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.525877471\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.525877471\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 + 2.42T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 0.341T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−10.07457433054508440845783199681, −9.293236970704672338718243667949, −8.759020588381798465208295632824, −8.108422552561443663373375688661, −6.69187263475088716660213028012, −6.08110690534179095771598775188, −4.49450432447704243393148714714, −3.60564484787814699608664502623, −3.31549828556117340819543001678, −1.68826769359262613429836749140,
1.68826769359262613429836749140, 3.31549828556117340819543001678, 3.60564484787814699608664502623, 4.49450432447704243393148714714, 6.08110690534179095771598775188, 6.69187263475088716660213028012, 8.108422552561443663373375688661, 8.759020588381798465208295632824, 9.293236970704672338718243667949, 10.07457433054508440845783199681