L(s) = 1 | + 1.92·2-s + 2.13·3-s + 1.70·4-s − 3.67·5-s + 4.10·6-s − 3.13·7-s − 0.571·8-s + 1.54·9-s − 7.07·10-s + 11-s + 3.63·12-s − 6.02·14-s − 7.84·15-s − 4.50·16-s + 3.22·17-s + 2.97·18-s − 3.26·19-s − 6.26·20-s − 6.67·21-s + 1.92·22-s − 8.55·23-s − 1.21·24-s + 8.52·25-s − 3.10·27-s − 5.33·28-s + 0.278·29-s − 15.0·30-s + ⋯ |
L(s) = 1 | + 1.36·2-s + 1.23·3-s + 0.851·4-s − 1.64·5-s + 1.67·6-s − 1.18·7-s − 0.201·8-s + 0.515·9-s − 2.23·10-s + 0.301·11-s + 1.04·12-s − 1.61·14-s − 2.02·15-s − 1.12·16-s + 0.783·17-s + 0.701·18-s − 0.749·19-s − 1.40·20-s − 1.45·21-s + 0.410·22-s − 1.78·23-s − 0.248·24-s + 1.70·25-s − 0.596·27-s − 1.00·28-s + 0.0516·29-s − 2.75·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 8.55T + 23T^{2} \) |
| 29 | \( 1 - 0.278T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 - 0.137T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 2.77T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 + 8.54T + 79T^{2} \) |
| 83 | \( 1 + 0.0460T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−8.533449760660486678523767288240, −8.196654310608500666526021484255, −7.15216183520786865174854075803, −6.49961858702078854421027148980, −5.46678099670926821849446660450, −4.24799476341591778290082718656, −3.68776561001437932814641571554, −3.35131443325711520681954964269, −2.35577851091326134290065297043, 0,
2.35577851091326134290065297043, 3.35131443325711520681954964269, 3.68776561001437932814641571554, 4.24799476341591778290082718656, 5.46678099670926821849446660450, 6.49961858702078854421027148980, 7.15216183520786865174854075803, 8.196654310608500666526021484255, 8.533449760660486678523767288240