| L(s) = 1 | − 1.45·2-s − 3.18·3-s + 0.119·4-s − 3.80·5-s + 4.64·6-s − 2.73·7-s + 2.73·8-s + 7.15·9-s + 5.53·10-s − 11-s − 0.381·12-s + 5.32·13-s + 3.98·14-s + 12.1·15-s − 4.22·16-s − 17-s − 10.4·18-s − 0.742·19-s − 0.454·20-s + 8.73·21-s + 1.45·22-s − 5.03·23-s − 8.72·24-s + 9.44·25-s − 7.75·26-s − 13.2·27-s − 0.327·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 1.84·3-s + 0.0597·4-s − 1.69·5-s + 1.89·6-s − 1.03·7-s + 0.967·8-s + 2.38·9-s + 1.74·10-s − 0.301·11-s − 0.109·12-s + 1.47·13-s + 1.06·14-s + 3.12·15-s − 1.05·16-s − 0.242·17-s − 2.45·18-s − 0.170·19-s − 0.101·20-s + 1.90·21-s + 0.310·22-s − 1.05·23-s − 1.78·24-s + 1.88·25-s − 1.52·26-s − 2.55·27-s − 0.0619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 19 | \( 1 + 0.742T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 0.331T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.05T + 97T^{2} \) |
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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
−7.44824841644493289239861866026, −6.95877803832255456512097762176, −6.13279408863840887243768997088, −5.63354799312480468565539487299, −4.48410724582281542832824186110, −4.11995825102732347130754822954, −3.40479567928715988975647713725, −1.61061972492598375082006824707, −0.57870811581231545880548778706, 0,
0.57870811581231545880548778706, 1.61061972492598375082006824707, 3.40479567928715988975647713725, 4.11995825102732347130754822954, 4.48410724582281542832824186110, 5.63354799312480468565539487299, 6.13279408863840887243768997088, 6.95877803832255456512097762176, 7.44824841644493289239861866026
