| L(s) = 1 | + 0.601·2-s + 0.879·3-s − 1.63·4-s − 5-s + 0.528·6-s − 7-s − 2.18·8-s − 2.22·9-s − 0.601·10-s − 1.99·11-s − 1.44·12-s + 4.61·13-s − 0.601·14-s − 0.879·15-s + 1.96·16-s + 2.33·17-s − 1.33·18-s − 1.48·19-s + 1.63·20-s − 0.879·21-s − 1.19·22-s + 6.09·23-s − 1.92·24-s + 25-s + 2.77·26-s − 4.59·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 0.425·2-s + 0.507·3-s − 0.819·4-s − 0.447·5-s + 0.215·6-s − 0.377·7-s − 0.773·8-s − 0.742·9-s − 0.190·10-s − 0.600·11-s − 0.415·12-s + 1.27·13-s − 0.160·14-s − 0.227·15-s + 0.490·16-s + 0.566·17-s − 0.315·18-s − 0.341·19-s + 0.366·20-s − 0.191·21-s − 0.255·22-s + 1.26·23-s − 0.392·24-s + 0.200·25-s + 0.544·26-s − 0.884·27-s + 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 0.601T + 2T^{2} \) |
| 3 | \( 1 - 0.879T + 3T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 2.33T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 - 0.177T + 47T^{2} \) |
| 53 | \( 1 - 0.337T + 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 8.69T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 - 3.81T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.68089456727834149970335821178, −6.70108858233698181742975496122, −5.97466008783956546904591029736, −5.34125655159215196992952467778, −4.64553845591687737808549825773, −3.70174071890402763946558299335, −3.31081663248480930907450010573, −2.60501359875526140947001789969, −1.13611593823274061064487292562, 0,
1.13611593823274061064487292562, 2.60501359875526140947001789969, 3.31081663248480930907450010573, 3.70174071890402763946558299335, 4.64553845591687737808549825773, 5.34125655159215196992952467778, 5.97466008783956546904591029736, 6.70108858233698181742975496122, 7.68089456727834149970335821178
