| L(s) = 1 | + 1.04·2-s − 0.908·4-s + 2.79·5-s − 4.32·7-s − 3.03·8-s + 2.92·10-s − 11-s + 2.42·13-s − 4.52·14-s − 1.35·16-s − 1.27·17-s + 5.47·19-s − 2.54·20-s − 1.04·22-s + 4.49·23-s + 2.83·25-s + 2.52·26-s + 3.93·28-s − 3.83·29-s + 0.921·31-s + 4.65·32-s − 1.33·34-s − 12.1·35-s − 7.51·37-s + 5.71·38-s − 8.50·40-s + 4.82·41-s + ⋯ |
| L(s) = 1 | + 0.738·2-s − 0.454·4-s + 1.25·5-s − 1.63·7-s − 1.07·8-s + 0.924·10-s − 0.301·11-s + 0.671·13-s − 1.20·14-s − 0.339·16-s − 0.309·17-s + 1.25·19-s − 0.568·20-s − 0.222·22-s + 0.936·23-s + 0.566·25-s + 0.495·26-s + 0.743·28-s − 0.712·29-s + 0.165·31-s + 0.823·32-s − 0.228·34-s − 2.04·35-s − 1.23·37-s + 0.927·38-s − 1.34·40-s + 0.754·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 4.49T + 23T^{2} \) |
| 29 | \( 1 + 3.83T + 29T^{2} \) |
| 31 | \( 1 - 0.921T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 + 9.03T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.51954896844100602872642165300, −6.71996809827781189827037916744, −6.08408077668340085556855029843, −5.63933529641240678513676523808, −5.00199147719673173541905642133, −3.95946061137148459425030164654, −3.18163189991011525617953036407, −2.73936276637477138154366358030, −1.37388113109033233065702147126, 0,
1.37388113109033233065702147126, 2.73936276637477138154366358030, 3.18163189991011525617953036407, 3.95946061137148459425030164654, 5.00199147719673173541905642133, 5.63933529641240678513676523808, 6.08408077668340085556855029843, 6.71996809827781189827037916744, 7.51954896844100602872642165300
