L(s) = 1 | − 2.57·2-s − 0.514·3-s + 4.63·4-s + 5-s + 1.32·6-s − 6.79·8-s − 2.73·9-s − 2.57·10-s + 11-s − 2.38·12-s − 1.97·13-s − 0.514·15-s + 8.23·16-s − 2.38·17-s + 7.04·18-s + 4.54·19-s + 4.63·20-s − 2.57·22-s − 2.71·23-s + 3.49·24-s + 25-s + 5.07·26-s + 2.95·27-s + 1.93·29-s + 1.32·30-s − 3.59·31-s − 7.62·32-s + ⋯ |
L(s) = 1 | − 1.82·2-s − 0.297·3-s + 2.31·4-s + 0.447·5-s + 0.541·6-s − 2.40·8-s − 0.911·9-s − 0.814·10-s + 0.301·11-s − 0.689·12-s − 0.546·13-s − 0.132·15-s + 2.05·16-s − 0.579·17-s + 1.66·18-s + 1.04·19-s + 1.03·20-s − 0.549·22-s − 0.566·23-s + 0.714·24-s + 0.200·25-s + 0.995·26-s + 0.568·27-s + 0.359·29-s + 0.242·30-s − 0.644·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.514T + 3T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 - 6.13T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 + 7.44T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 + 0.932T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 + 0.820T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
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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
−8.644402207979783441275351464240, −7.76668166036400914776528507826, −7.26877816855387910952917917505, −6.23269851443358449117778614150, −5.85373539030941638397826865277, −4.63181559045537139996133038862, −3.07302699131390671506261534962, −2.31869808116812089850067037479, −1.21451284079745669042707566071, 0,
1.21451284079745669042707566071, 2.31869808116812089850067037479, 3.07302699131390671506261534962, 4.63181559045537139996133038862, 5.85373539030941638397826865277, 6.23269851443358449117778614150, 7.26877816855387910952917917505, 7.76668166036400914776528507826, 8.644402207979783441275351464240