| L(s) = 1 | + 3.26·3-s − 5-s − 1.11·7-s + 7.63·9-s + 3.11·11-s − 1.11·13-s − 3.26·15-s − 0.377·17-s + 6.11·19-s − 3.63·21-s − 7.63·23-s + 25-s + 15.1·27-s + 5.63·29-s − 31-s + 10.1·33-s + 1.11·35-s + 4.14·37-s − 3.63·39-s − 8.04·41-s − 11.4·43-s − 7.63·45-s − 2.52·47-s − 5.75·49-s − 1.23·51-s + 2.01·53-s − 3.11·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s − 0.447·5-s − 0.421·7-s + 2.54·9-s + 0.939·11-s − 0.309·13-s − 0.842·15-s − 0.0915·17-s + 1.40·19-s − 0.794·21-s − 1.59·23-s + 0.200·25-s + 2.91·27-s + 1.04·29-s − 0.179·31-s + 1.76·33-s + 0.188·35-s + 0.681·37-s − 0.582·39-s − 1.25·41-s − 1.74·43-s − 1.13·45-s − 0.368·47-s − 0.822·49-s − 0.172·51-s + 0.277·53-s − 0.420·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.603090939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.603090939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 + 0.377T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 + 7.63T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 5.72T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 + 0.755T + 97T^{2} \) |
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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
−10.09999034479511322693100122671, −9.701884128885110357784516844985, −8.804250593409703629129334147213, −8.093569712669832246212012715761, −7.33559322371250740489207306901, −6.43668514439369478537176311986, −4.67480273482346826378612563843, −3.67678324889039567497890175119, −3.00455886170072183236959753320, −1.64053283404703884584644514616,
1.64053283404703884584644514616, 3.00455886170072183236959753320, 3.67678324889039567497890175119, 4.67480273482346826378612563843, 6.43668514439369478537176311986, 7.33559322371250740489207306901, 8.093569712669832246212012715761, 8.804250593409703629129334147213, 9.701884128885110357784516844985, 10.09999034479511322693100122671
