| L(s) = 1 | − 2.58·2-s + 4.68·4-s − 5-s − 4.27·7-s − 6.95·8-s + 2.58·10-s + 4.38·11-s − 13-s + 11.0·14-s + 8.60·16-s + 0.619·17-s − 2.61·19-s − 4.68·20-s − 11.3·22-s + 5.42·23-s + 25-s + 2.58·26-s − 20.0·28-s + 6.95·29-s − 6.88·31-s − 8.34·32-s − 1.60·34-s + 4.27·35-s − 4.02·37-s + 6.75·38-s + 6.95·40-s − 4.07·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.34·4-s − 0.447·5-s − 1.61·7-s − 2.45·8-s + 0.817·10-s + 1.32·11-s − 0.277·13-s + 2.95·14-s + 2.15·16-s + 0.150·17-s − 0.599·19-s − 1.04·20-s − 2.41·22-s + 1.13·23-s + 0.200·25-s + 0.507·26-s − 3.78·28-s + 1.29·29-s − 1.23·31-s − 1.47·32-s − 0.274·34-s + 0.722·35-s − 0.661·37-s + 1.09·38-s + 1.09·40-s − 0.636·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 17 | \( 1 - 0.619T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 - 6.95T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 0.405T + 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.07T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 9.63T + 97T^{2} \) |
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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
−8.044528520337738562713136085454, −7.07933474879407819747397434129, −6.62969720876583488488713636490, −6.36438861349281697290891002958, −5.02383453230043179014270892511, −3.65454168482907342216616238553, −3.17354011399444076331557011474, −2.07474842953010138455511116457, −0.975786282539215729543883448731, 0,
0.975786282539215729543883448731, 2.07474842953010138455511116457, 3.17354011399444076331557011474, 3.65454168482907342216616238553, 5.02383453230043179014270892511, 6.36438861349281697290891002958, 6.62969720876583488488713636490, 7.07933474879407819747397434129, 8.044528520337738562713136085454
