L(s) = 1 | + 3.64·5-s + 7-s + 2·11-s − 4·13-s − 0.354·17-s + 19-s − 6·23-s + 8.29·25-s + 5.64·29-s + 1.29·31-s + 3.64·35-s + 5.29·37-s − 8.58·41-s − 2·43-s + 2.35·47-s + 49-s − 8.93·53-s + 7.29·55-s + 14.5·59-s + 2.70·61-s − 14.5·65-s + 14.5·67-s + 0.354·71-s − 9.29·73-s + 2·77-s + 11.2·79-s + 16.9·83-s + ⋯ |
L(s) = 1 | + 1.63·5-s + 0.377·7-s + 0.603·11-s − 1.10·13-s − 0.0859·17-s + 0.229·19-s − 1.25·23-s + 1.65·25-s + 1.04·29-s + 0.231·31-s + 0.616·35-s + 0.869·37-s − 1.34·41-s − 0.304·43-s + 0.343·47-s + 0.142·49-s − 1.22·53-s + 0.983·55-s + 1.89·59-s + 0.346·61-s − 1.80·65-s + 1.78·67-s + 0.0420·71-s − 1.08·73-s + 0.227·77-s + 1.27·79-s + 1.85·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.161612286\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.161612286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 0.354T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 8.58T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 0.354T + 71T^{2} \) |
| 73 | \( 1 + 9.29T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.74697340580436266784098596421, −6.69716885087954619688666436482, −6.46805534521897192994987711042, −5.58419501201243826331107918807, −5.05907130540039012820748234041, −4.37129029911857361233485314990, −3.32028826377658348341060409570, −2.30829165602951335449140978382, −1.93922235124099235517751480010, −0.860121756746707250230110630271,
0.860121756746707250230110630271, 1.93922235124099235517751480010, 2.30829165602951335449140978382, 3.32028826377658348341060409570, 4.37129029911857361233485314990, 5.05907130540039012820748234041, 5.58419501201243826331107918807, 6.46805534521897192994987711042, 6.69716885087954619688666436482, 7.74697340580436266784098596421