| L(s) = 1 | + 20.7·3-s + 25·5-s + 35.2·7-s + 186.·9-s − 7.33·11-s + 619.·13-s + 518.·15-s + 959.·17-s − 309.·19-s + 731.·21-s + 2.46e3·23-s + 625·25-s − 1.16e3·27-s − 1.28e3·29-s + 7.09e3·31-s − 152.·33-s + 881.·35-s − 6.10e3·37-s + 1.28e4·39-s − 1.88e4·41-s + 3.14e3·43-s + 4.67e3·45-s + 2.05e4·47-s − 1.55e4·49-s + 1.98e4·51-s + 3.37e4·53-s − 183.·55-s + ⋯ |
| L(s) = 1 | + 1.33·3-s + 0.447·5-s + 0.272·7-s + 0.768·9-s − 0.0182·11-s + 1.01·13-s + 0.594·15-s + 0.805·17-s − 0.196·19-s + 0.361·21-s + 0.972·23-s + 0.200·25-s − 0.307·27-s − 0.283·29-s + 1.32·31-s − 0.0242·33-s + 0.121·35-s − 0.732·37-s + 1.35·39-s − 1.74·41-s + 0.259·43-s + 0.343·45-s + 1.35·47-s − 0.925·49-s + 1.07·51-s + 1.64·53-s − 0.00817·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.629032205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.629032205\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 20.7T + 243T^{2} \) |
| 7 | \( 1 - 35.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 7.33T + 1.61e5T^{2} \) |
| 13 | \( 1 - 619.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 959.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 309.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.10e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.54e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.76e5T + 8.58e9T^{2} \) |
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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
−12.10334818284488564526303324012, −10.81809875389543889540661304757, −9.769666204051376942254857200471, −8.762616218540841382037509222076, −8.121813769532523070459085255303, −6.83374691369088593480486999847, −5.40962893907297669090338215654, −3.81535126494829123129434427721, −2.70963162136766918405162495655, −1.34742039953208690732897125716,
1.34742039953208690732897125716, 2.70963162136766918405162495655, 3.81535126494829123129434427721, 5.40962893907297669090338215654, 6.83374691369088593480486999847, 8.121813769532523070459085255303, 8.762616218540841382037509222076, 9.769666204051376942254857200471, 10.81809875389543889540661304757, 12.10334818284488564526303324012
