| L(s) = 1 | + 2.61·3-s + 1.63·5-s − 3.46·7-s + 3.85·9-s + 1.65·11-s + 2.82·13-s + 4.27·15-s + 6.37·17-s + 3.98·19-s − 9.07·21-s + 2.04·23-s − 2.33·25-s + 2.25·27-s + 0.0143·29-s − 7.31·31-s + 4.32·33-s − 5.65·35-s + 4.31·37-s + 7.38·39-s + 10.1·41-s − 1.72·43-s + 6.29·45-s + 47-s + 5.00·49-s + 16.6·51-s − 5.63·53-s + 2.69·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 0.729·5-s − 1.30·7-s + 1.28·9-s + 0.498·11-s + 0.782·13-s + 1.10·15-s + 1.54·17-s + 0.915·19-s − 1.98·21-s + 0.426·23-s − 0.467·25-s + 0.433·27-s + 0.00265·29-s − 1.31·31-s + 0.753·33-s − 0.955·35-s + 0.709·37-s + 1.18·39-s + 1.58·41-s − 0.263·43-s + 0.938·45-s + 0.145·47-s + 0.715·49-s + 2.33·51-s − 0.774·53-s + 0.363·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.112581659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.112581659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 - 0.0143T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 1.84T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 - 1.78T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 7.77T + 83T^{2} \) |
| 89 | \( 1 + 0.300T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−7.941223279142175896493875410421, −7.64314801551486321573190416590, −6.67424437800044379841272303065, −6.02161792413389952924448437690, −5.36757856718935453206378080663, −4.02987376817940086269359414086, −3.38755376279136366331631356957, −3.01111671868481752728376008857, −1.99318959740066318889323145823, −1.04905942433937012868822209808,
1.04905942433937012868822209808, 1.99318959740066318889323145823, 3.01111671868481752728376008857, 3.38755376279136366331631356957, 4.02987376817940086269359414086, 5.36757856718935453206378080663, 6.02161792413389952924448437690, 6.67424437800044379841272303065, 7.64314801551486321573190416590, 7.941223279142175896493875410421
