| L(s) = 1 | − 2.64·2-s − 3-s + 4.97·4-s − 5-s + 2.64·6-s − 7.84·8-s + 9-s + 2.64·10-s − 11-s − 4.97·12-s + 1.46·13-s + 15-s + 10.7·16-s − 1.32·17-s − 2.64·18-s + 1.90·19-s − 4.97·20-s + 2.64·22-s + 6.10·23-s + 7.84·24-s + 25-s − 3.87·26-s − 27-s − 5.36·29-s − 2.64·30-s + 5.25·31-s − 12.7·32-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.48·4-s − 0.447·5-s + 1.07·6-s − 2.77·8-s + 0.333·9-s + 0.834·10-s − 0.301·11-s − 1.43·12-s + 0.407·13-s + 0.258·15-s + 2.69·16-s − 0.320·17-s − 0.622·18-s + 0.437·19-s − 1.11·20-s + 0.562·22-s + 1.27·23-s + 1.60·24-s + 0.200·25-s − 0.760·26-s − 0.192·27-s − 0.996·29-s − 0.482·30-s + 0.944·31-s − 2.25·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 0.910T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 - 4.43T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.64797863506119269066629942981, −6.80390893098024155995002208334, −6.64891777512335576633481363762, −5.58080247787776673220830012332, −4.86681345345414567062672765282, −3.62992697646274969937681428564, −2.83349322315087193861480915190, −1.79150579113107842512943697200, −0.955415594434073758446691344980, 0,
0.955415594434073758446691344980, 1.79150579113107842512943697200, 2.83349322315087193861480915190, 3.62992697646274969937681428564, 4.86681345345414567062672765282, 5.58080247787776673220830012332, 6.64891777512335576633481363762, 6.80390893098024155995002208334, 7.64797863506119269066629942981
