| L(s) = 1 | − 2-s + 1.75·3-s + 4-s + 5-s − 1.75·6-s + 2.75·7-s − 8-s + 0.0861·9-s − 10-s − 11-s + 1.75·12-s − 3.85·13-s − 2.75·14-s + 1.75·15-s + 16-s − 0.956·17-s − 0.0861·18-s − 8.01·19-s + 20-s + 4.84·21-s + 22-s − 0.472·23-s − 1.75·24-s + 25-s + 3.85·26-s − 5.11·27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.01·3-s + 0.5·4-s + 0.447·5-s − 0.717·6-s + 1.04·7-s − 0.353·8-s + 0.0287·9-s − 0.316·10-s − 0.301·11-s + 0.507·12-s − 1.07·13-s − 0.737·14-s + 0.453·15-s + 0.250·16-s − 0.231·17-s − 0.0202·18-s − 1.83·19-s + 0.223·20-s + 1.05·21-s + 0.213·22-s − 0.0984·23-s − 0.358·24-s + 0.200·25-s + 0.756·26-s − 0.985·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 + 0.956T + 17T^{2} \) |
| 19 | \( 1 + 8.01T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 - 1.22T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.82646331968931429817879202786, −6.97482497762862862581854735714, −6.29750936424429399587841798555, −5.36544638442544924939601829285, −4.65652354024704926645274750435, −3.82435792070440542780292260272, −2.56750448271013545249926447152, −2.37072899152346346691351798085, −1.49336294265847662128090630224, 0,
1.49336294265847662128090630224, 2.37072899152346346691351798085, 2.56750448271013545249926447152, 3.82435792070440542780292260272, 4.65652354024704926645274750435, 5.36544638442544924939601829285, 6.29750936424429399587841798555, 6.97482497762862862581854735714, 7.82646331968931429817879202786
