| L(s) = 1 | − 16.8·3-s + 47.4·5-s − 67.4·7-s + 42.2·9-s − 81.3·11-s + 1.05e3·13-s − 801.·15-s + 251.·17-s − 1.61e3·19-s + 1.13e3·21-s + 32.7·23-s − 872.·25-s + 3.39e3·27-s − 2.58e3·29-s + 7.20e3·31-s + 1.37e3·33-s − 3.20e3·35-s − 6.17e3·37-s − 1.78e4·39-s + 1.55e4·41-s − 1.84e3·43-s + 2.00e3·45-s + 1.69e3·47-s − 1.22e4·49-s − 4.24e3·51-s − 2.56e4·53-s − 3.86e3·55-s + ⋯ |
| L(s) = 1 | − 1.08·3-s + 0.849·5-s − 0.520·7-s + 0.173·9-s − 0.202·11-s + 1.73·13-s − 0.919·15-s + 0.210·17-s − 1.02·19-s + 0.563·21-s + 0.0129·23-s − 0.279·25-s + 0.895·27-s − 0.570·29-s + 1.34·31-s + 0.219·33-s − 0.441·35-s − 0.741·37-s − 1.88·39-s + 1.44·41-s − 0.152·43-s + 0.147·45-s + 0.111·47-s − 0.729·49-s − 0.228·51-s − 1.25·53-s − 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.392417133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392417133\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 3 | \( 1 + 16.8T + 243T^{2} \) |
| 5 | \( 1 - 47.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 67.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 81.3T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 251.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 32.7T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.55e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.20e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.47e5T + 8.58e9T^{2} \) |
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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
−9.855105017681873830244910348180, −8.938269617005337826652094160848, −8.037972672456992679836642824765, −6.55419214905611900576247376318, −6.16583248028176961358429973963, −5.50065973294948712000809873783, −4.32448222822699516067872240714, −3.09876683216778981539007243232, −1.73340381601177078228900649807, −0.59534091218881285753932318848,
0.59534091218881285753932318848, 1.73340381601177078228900649807, 3.09876683216778981539007243232, 4.32448222822699516067872240714, 5.50065973294948712000809873783, 6.16583248028176961358429973963, 6.55419214905611900576247376318, 8.037972672456992679836642824765, 8.938269617005337826652094160848, 9.855105017681873830244910348180
