| L(s) = 1 | − 2.43·2-s − 0.839·3-s + 3.91·4-s − 3.64·5-s + 2.04·6-s − 2.93·7-s − 4.66·8-s − 2.29·9-s + 8.86·10-s − 1.95·11-s − 3.29·12-s + 13-s + 7.14·14-s + 3.06·15-s + 3.51·16-s − 6.09·17-s + 5.58·18-s + 6.19·19-s − 14.2·20-s + 2.46·21-s + 4.74·22-s − 5.35·23-s + 3.91·24-s + 8.28·25-s − 2.43·26-s + 4.44·27-s − 11.5·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.484·3-s + 1.95·4-s − 1.63·5-s + 0.834·6-s − 1.11·7-s − 1.64·8-s − 0.764·9-s + 2.80·10-s − 0.588·11-s − 0.949·12-s + 0.277·13-s + 1.90·14-s + 0.790·15-s + 0.878·16-s − 1.47·17-s + 1.31·18-s + 1.42·19-s − 3.19·20-s + 0.538·21-s + 1.01·22-s − 1.11·23-s + 0.799·24-s + 1.65·25-s − 0.477·26-s + 0.855·27-s − 2.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.839T + 3T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 5.35T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 + 2.60T + 31T^{2} \) |
| 37 | \( 1 - 0.220T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.488T + 47T^{2} \) |
| 53 | \( 1 + 2.57T + 53T^{2} \) |
| 59 | \( 1 - 0.288T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.69367144720384633924675980418, −6.90819636576464398346291985467, −6.57855403123584020367875381657, −5.62607712976339308013137360529, −4.68441845680191908732238582485, −3.55022497183491702204257833718, −3.08358507384782000369621709058, −1.98136524128240076373646419179, −0.53310974365775914651579631114, 0,
0.53310974365775914651579631114, 1.98136524128240076373646419179, 3.08358507384782000369621709058, 3.55022497183491702204257833718, 4.68441845680191908732238582485, 5.62607712976339308013137360529, 6.57855403123584020367875381657, 6.90819636576464398346291985467, 7.69367144720384633924675980418
