| L(s) = 1 | − 64.1·2-s + 26.5·3-s + 2.07e3·4-s − 1.74e3·5-s − 1.70e3·6-s + 3.90e4·7-s − 1.61e3·8-s − 1.76e5·9-s + 1.12e5·10-s + 8.15e5·11-s + 5.50e4·12-s − 9.39e4·13-s − 2.50e6·14-s − 4.64e4·15-s − 4.14e6·16-s + 7.89e5·17-s + 1.13e7·18-s − 9.66e6·19-s − 3.62e6·20-s + 1.03e6·21-s − 5.23e7·22-s − 3.17e7·23-s − 4.29e4·24-s − 4.57e7·25-s + 6.03e6·26-s − 9.38e6·27-s + 8.09e7·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.0630·3-s + 1.01·4-s − 0.250·5-s − 0.0894·6-s + 0.878·7-s − 0.0174·8-s − 0.996·9-s + 0.355·10-s + 1.52·11-s + 0.0638·12-s − 0.0701·13-s − 1.24·14-s − 0.0157·15-s − 0.987·16-s + 0.134·17-s + 1.41·18-s − 0.895·19-s − 0.253·20-s + 0.0553·21-s − 2.16·22-s − 1.02·23-s − 0.00110·24-s − 0.937·25-s + 0.0995·26-s − 0.125·27-s + 0.888·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.8971667285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8971667285\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + 1.15e8T \) |
| good | 2 | \( 1 + 64.1T + 2.04e3T^{2} \) |
| 3 | \( 1 - 26.5T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.74e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.90e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.15e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.39e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.89e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.66e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.17e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.78e6T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.30e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.07e8T + 1.77e17T^{2} \) |
| 43 | \( 1 - 1.64e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.50e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.61e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.02e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.57e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.44e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.14e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.56e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.38e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.68e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.13e10T + 7.15e21T^{2} \) |
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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
−13.92443004667841001282845059609, −11.85736221543361275887806318364, −11.15965089254256848119842566917, −9.739557855761928292974673464420, −8.621907672624321609308471170477, −7.86320103592960614476782780936, −6.27508201866023896874997081027, −4.23839592328140768679901152917, −2.07693620942142325193109314345, −0.73810095664879778616036983871,
0.73810095664879778616036983871, 2.07693620942142325193109314345, 4.23839592328140768679901152917, 6.27508201866023896874997081027, 7.86320103592960614476782780936, 8.621907672624321609308471170477, 9.739557855761928292974673464420, 11.15965089254256848119842566917, 11.85736221543361275887806318364, 13.92443004667841001282845059609
