L(s) = 1 | − 2.28·2-s − 3.30·3-s + 3.24·4-s + 5-s + 7.57·6-s − 2.93·7-s − 2.84·8-s + 7.93·9-s − 2.28·10-s − 0.577·11-s − 10.7·12-s + 0.670·13-s + 6.73·14-s − 3.30·15-s + 0.0353·16-s − 0.351·17-s − 18.1·18-s + 3.24·20-s + 9.72·21-s + 1.32·22-s − 4.12·23-s + 9.42·24-s + 25-s − 1.53·26-s − 16.3·27-s − 9.53·28-s − 3.00·29-s + ⋯ |
L(s) = 1 | − 1.61·2-s − 1.90·3-s + 1.62·4-s + 0.447·5-s + 3.09·6-s − 1.11·7-s − 1.00·8-s + 2.64·9-s − 0.724·10-s − 0.174·11-s − 3.09·12-s + 0.185·13-s + 1.79·14-s − 0.853·15-s + 0.00883·16-s − 0.0853·17-s − 4.28·18-s + 0.725·20-s + 2.12·21-s + 0.281·22-s − 0.859·23-s + 1.92·24-s + 0.200·25-s − 0.301·26-s − 3.14·27-s − 1.80·28-s − 0.558·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 0.577T + 11T^{2} \) |
| 13 | \( 1 - 0.670T + 13T^{2} \) |
| 17 | \( 1 + 0.351T + 17T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 31 | \( 1 + 0.297T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 + 6.59T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 + 0.610T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 + 1.54T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.280135904647852129804586130646, −8.054310480770019861185401325198, −7.18467046964110763637283054012, −6.50506092381414021091801073233, −6.04507786215266693865583671892, −5.13117426796273546781231478670, −3.91170209701622331586888212965, −2.21310454043646180539177097685, −0.992130739114886778620585115539, 0,
0.992130739114886778620585115539, 2.21310454043646180539177097685, 3.91170209701622331586888212965, 5.13117426796273546781231478670, 6.04507786215266693865583671892, 6.50506092381414021091801073233, 7.18467046964110763637283054012, 8.054310480770019861185401325198, 9.280135904647852129804586130646