| L(s) = 1 | + 12.5·7-s − 164.·11-s − 849.·13-s − 1.60e3·17-s − 446.·19-s + 802.·23-s − 4.47e3·29-s + 5.20e3·31-s − 8.24e3·37-s + 6.25e3·41-s + 1.13e4·43-s + 2.98e4·47-s − 1.66e4·49-s + 9.19e3·53-s − 9.94e3·59-s + 4.15e4·61-s − 4.90e4·67-s + 4.49e4·71-s − 6.56e4·73-s − 2.07e3·77-s − 6.71e4·79-s + 5.48e4·83-s + 5.39e4·89-s − 1.06e4·91-s − 1.00e5·97-s − 3.51e4·101-s + 2.62e4·103-s + ⋯ |
| L(s) = 1 | + 0.0970·7-s − 0.410·11-s − 1.39·13-s − 1.34·17-s − 0.283·19-s + 0.316·23-s − 0.988·29-s + 0.972·31-s − 0.990·37-s + 0.580·41-s + 0.933·43-s + 1.97·47-s − 0.990·49-s + 0.449·53-s − 0.372·59-s + 1.42·61-s − 1.33·67-s + 1.05·71-s − 1.44·73-s − 0.0398·77-s − 1.21·79-s + 0.874·83-s + 0.722·89-s − 0.135·91-s − 1.08·97-s − 0.343·101-s + 0.243·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.309928267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309928267\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 12.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 164.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 849.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.60e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 446.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 802.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.98e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.19e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.94e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
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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
−9.322838973214231493514482240071, −8.615668702766901899040863529176, −7.55601164187925299004294566675, −6.96255999046426698023266899966, −5.86936404860790991495472785003, −4.90832263450580867861844099516, −4.15134240577473348277877970645, −2.77005786694108818730106032678, −2.01433632391251129021888671654, −0.48574405470212506600411229526,
0.48574405470212506600411229526, 2.01433632391251129021888671654, 2.77005786694108818730106032678, 4.15134240577473348277877970645, 4.90832263450580867861844099516, 5.86936404860790991495472785003, 6.96255999046426698023266899966, 7.55601164187925299004294566675, 8.615668702766901899040863529176, 9.322838973214231493514482240071
