L(s) = 1 | − 0.248·2-s − 2.62·3-s − 1.93·4-s + 3.83·5-s + 0.650·6-s + 2.64·7-s + 0.977·8-s + 3.87·9-s − 0.951·10-s − 5.27·11-s + 5.08·12-s + 1.02·13-s − 0.655·14-s − 10.0·15-s + 3.63·16-s + 17-s − 0.961·18-s − 4.44·19-s − 7.43·20-s − 6.92·21-s + 1.30·22-s − 1.34·23-s − 2.56·24-s + 9.72·25-s − 0.253·26-s − 2.29·27-s − 5.11·28-s + ⋯ |
L(s) = 1 | − 0.175·2-s − 1.51·3-s − 0.969·4-s + 1.71·5-s + 0.265·6-s + 0.997·7-s + 0.345·8-s + 1.29·9-s − 0.301·10-s − 1.59·11-s + 1.46·12-s + 0.283·13-s − 0.175·14-s − 2.59·15-s + 0.908·16-s + 0.242·17-s − 0.226·18-s − 1.01·19-s − 1.66·20-s − 1.51·21-s + 0.279·22-s − 0.280·23-s − 0.522·24-s + 1.94·25-s − 0.0497·26-s − 0.441·27-s − 0.967·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9208980572\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9208980572\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 0.248T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.23T + 53T^{2} \) |
| 59 | \( 1 - 9.75T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 - 0.0170T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−10.36116571034444988487055522203, −9.920460723789474573845701303997, −8.711437394235662699660609791242, −7.929356853287680015426603701316, −6.52212803181930747577155848885, −5.67960533919205198119229154170, −5.18352234859819266305812309701, −4.50890698309273185357118532028, −2.32156410186170697615815719694, −0.926507428058740749370233676131,
0.926507428058740749370233676131, 2.32156410186170697615815719694, 4.50890698309273185357118532028, 5.18352234859819266305812309701, 5.67960533919205198119229154170, 6.52212803181930747577155848885, 7.929356853287680015426603701316, 8.711437394235662699660609791242, 9.920460723789474573845701303997, 10.36116571034444988487055522203