| L(s) = 1 | + 0.220·2-s − 3-s − 1.95·4-s − 3.79·5-s − 0.220·6-s − 3.98·7-s − 0.872·8-s + 9-s − 0.836·10-s − 3.30·11-s + 1.95·12-s + 0.531·13-s − 0.880·14-s + 3.79·15-s + 3.70·16-s + 17-s + 0.220·18-s + 3.84·19-s + 7.39·20-s + 3.98·21-s − 0.729·22-s + 3.07·23-s + 0.872·24-s + 9.36·25-s + 0.117·26-s − 27-s + 7.78·28-s + ⋯ |
| L(s) = 1 | + 0.156·2-s − 0.577·3-s − 0.975·4-s − 1.69·5-s − 0.0901·6-s − 1.50·7-s − 0.308·8-s + 0.333·9-s − 0.264·10-s − 0.995·11-s + 0.563·12-s + 0.147·13-s − 0.235·14-s + 0.978·15-s + 0.927·16-s + 0.242·17-s + 0.0520·18-s + 0.882·19-s + 1.65·20-s + 0.870·21-s − 0.155·22-s + 0.640·23-s + 0.178·24-s + 1.87·25-s + 0.0230·26-s − 0.192·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.220T + 2T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 - 0.531T + 13T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 0.237T + 29T^{2} \) |
| 31 | \( 1 - 2.26T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.20T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 83 | \( 1 + 0.683T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−7.991603877962529742246882147772, −7.39160448678098557063480818103, −6.68273421810410782014018606131, −5.69156852758026956621010089787, −5.02598072156018885425542938330, −4.23937024468610947381842201466, −3.44003575589884693490421410058, −3.02341034513278627648077094748, −0.77474603154944400618317149250, 0,
0.77474603154944400618317149250, 3.02341034513278627648077094748, 3.44003575589884693490421410058, 4.23937024468610947381842201466, 5.02598072156018885425542938330, 5.69156852758026956621010089787, 6.68273421810410782014018606131, 7.39160448678098557063480818103, 7.991603877962529742246882147772
