L(s) = 1 | − 2.55·2-s + 3-s + 4.52·4-s − 2.76·5-s − 2.55·6-s − 7-s − 6.44·8-s + 9-s + 7.05·10-s − 5.11·11-s + 4.52·12-s + 1.44·13-s + 2.55·14-s − 2.76·15-s + 7.41·16-s + 2.55·17-s − 2.55·18-s + 2.24·19-s − 12.4·20-s − 21-s + 13.0·22-s − 0.475·23-s − 6.44·24-s + 2.62·25-s − 3.69·26-s + 27-s − 4.52·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 0.577·3-s + 2.26·4-s − 1.23·5-s − 1.04·6-s − 0.377·7-s − 2.27·8-s + 0.333·9-s + 2.22·10-s − 1.54·11-s + 1.30·12-s + 0.400·13-s + 0.682·14-s − 0.712·15-s + 1.85·16-s + 0.618·17-s − 0.602·18-s + 0.515·19-s − 2.79·20-s − 0.218·21-s + 2.78·22-s − 0.0991·23-s − 1.31·24-s + 0.524·25-s − 0.723·26-s + 0.192·27-s − 0.854·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 0.475T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 4.97T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 + 0.613T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 + 0.0375T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 + 6.35T + 71T^{2} \) |
| 73 | \( 1 - 9.02T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 - 3.68T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.60393732107773153769228464326, −7.54479412649844682156168922786, −6.43402601689197882014194676442, −5.65402892881226172089467007768, −4.54267495283016364644481298062, −3.49174356223353141065976143761, −2.93476615795673934534173811184, −2.08693115722404674858161744274, −0.913404129704081538507192064988, 0,
0.913404129704081538507192064988, 2.08693115722404674858161744274, 2.93476615795673934534173811184, 3.49174356223353141065976143761, 4.54267495283016364644481298062, 5.65402892881226172089467007768, 6.43402601689197882014194676442, 7.54479412649844682156168922786, 7.60393732107773153769228464326