| L(s) = 1 | − 8.25·2-s − 30.3·3-s + 36.0·4-s − 26.4·5-s + 250.·6-s − 114.·7-s − 33.5·8-s + 677.·9-s + 218.·10-s + 428.·11-s − 1.09e3·12-s − 862.·13-s + 948.·14-s + 801.·15-s − 877.·16-s − 1.33e3·17-s − 5.58e3·18-s − 1.16e3·19-s − 953.·20-s + 3.48e3·21-s − 3.53e3·22-s + 1.38e3·23-s + 1.01e3·24-s − 2.42e3·25-s + 7.11e3·26-s − 1.31e4·27-s − 4.14e3·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 1.94·3-s + 1.12·4-s − 0.472·5-s + 2.83·6-s − 0.887·7-s − 0.185·8-s + 2.78·9-s + 0.689·10-s + 1.06·11-s − 2.19·12-s − 1.41·13-s + 1.29·14-s + 0.919·15-s − 0.856·16-s − 1.11·17-s − 4.06·18-s − 0.742·19-s − 0.532·20-s + 1.72·21-s − 1.55·22-s + 0.546·23-s + 0.360·24-s − 0.776·25-s + 2.06·26-s − 3.47·27-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + 2.99e5T \) |
| good | 2 | \( 1 + 8.25T + 32T^{2} \) |
| 3 | \( 1 + 30.3T + 243T^{2} \) |
| 5 | \( 1 + 26.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 114.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 428.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 862.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.65e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.52e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.39e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.73e4T + 8.58e9T^{2} \) |
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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
−9.770307986988551707381325039358, −8.970468542813157308645249578209, −7.58086097030109451713823373042, −6.77357034015742180387086307698, −6.40465270549416764739611142524, −4.96072321127129189715628723947, −4.08304083569500062343204608952, −1.95960175168143355468533717188, −0.65624110715285019706264244095, 0,
0.65624110715285019706264244095, 1.95960175168143355468533717188, 4.08304083569500062343204608952, 4.96072321127129189715628723947, 6.40465270549416764739611142524, 6.77357034015742180387086307698, 7.58086097030109451713823373042, 8.970468542813157308645249578209, 9.770307986988551707381325039358
