L(s) = 1 | − 2.72·2-s + 5.42·4-s − 2.18·5-s − 9.33·8-s + 5.96·10-s + 1.04·11-s + 13-s + 14.5·16-s + 5.29·17-s + 0.756·19-s − 11.8·20-s − 2.85·22-s − 0.653·23-s − 0.216·25-s − 2.72·26-s + 3.10·29-s + 1.02·31-s − 21.1·32-s − 14.4·34-s − 10.8·37-s − 2.06·38-s + 20.4·40-s − 7.32·41-s + 0.887·43-s + 5.68·44-s + 1.78·46-s − 2.33·47-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 2.71·4-s − 0.978·5-s − 3.30·8-s + 1.88·10-s + 0.316·11-s + 0.277·13-s + 3.64·16-s + 1.28·17-s + 0.173·19-s − 2.65·20-s − 0.609·22-s − 0.136·23-s − 0.0432·25-s − 0.534·26-s + 0.576·29-s + 0.184·31-s − 3.73·32-s − 2.47·34-s − 1.79·37-s − 0.334·38-s + 3.22·40-s − 1.14·41-s + 0.135·43-s + 0.857·44-s + 0.262·46-s − 0.340·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 0.756T + 19T^{2} \) |
| 23 | \( 1 + 0.653T + 23T^{2} \) |
| 29 | \( 1 - 3.10T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 0.887T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 2.88T + 97T^{2} \) |
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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
−7.81489515883313391727522046871, −7.45488106024033908678744630441, −6.66939122227202175471943989501, −6.01988955708404303366429494581, −4.98717588152144745557689818327, −3.62006682495064211634775079893, −3.15080344957019487566401294103, −1.91095870364037716394963343251, −1.05866358140748898430662918406, 0,
1.05866358140748898430662918406, 1.91095870364037716394963343251, 3.15080344957019487566401294103, 3.62006682495064211634775079893, 4.98717588152144745557689818327, 6.01988955708404303366429494581, 6.66939122227202175471943989501, 7.45488106024033908678744630441, 7.81489515883313391727522046871