L(s) = 1 | + 2.74·2-s + 1.44·3-s + 5.51·4-s + 5-s + 3.97·6-s + 2.53·7-s + 9.63·8-s − 0.897·9-s + 2.74·10-s − 11-s + 7.99·12-s + 4.13·13-s + 6.94·14-s + 1.44·15-s + 15.3·16-s − 2.66·17-s − 2.46·18-s − 5.59·19-s + 5.51·20-s + 3.67·21-s − 2.74·22-s − 6.74·23-s + 13.9·24-s + 25-s + 11.3·26-s − 5.65·27-s + 13.9·28-s + ⋯ |
L(s) = 1 | + 1.93·2-s + 0.837·3-s + 2.75·4-s + 0.447·5-s + 1.62·6-s + 0.958·7-s + 3.40·8-s − 0.299·9-s + 0.866·10-s − 0.301·11-s + 2.30·12-s + 1.14·13-s + 1.85·14-s + 0.374·15-s + 3.84·16-s − 0.646·17-s − 0.580·18-s − 1.28·19-s + 1.23·20-s + 0.802·21-s − 0.584·22-s − 1.40·23-s + 2.85·24-s + 0.200·25-s + 2.22·26-s − 1.08·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.47351559\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.47351559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 8.29T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 - 7.01T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 79 | \( 1 + 7.99T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.316395592242562414120523053257, −7.59262829704971978565538093250, −6.73318042957305865685988907866, −5.84539042531481708987731249563, −5.59222006742685045216758070156, −4.41157120346419601977562021846, −4.02183649934167480758755760661, −3.10787040480254854172483580365, −2.16616466775836724020926018032, −1.79076488606077558170236415221,
1.79076488606077558170236415221, 2.16616466775836724020926018032, 3.10787040480254854172483580365, 4.02183649934167480758755760661, 4.41157120346419601977562021846, 5.59222006742685045216758070156, 5.84539042531481708987731249563, 6.73318042957305865685988907866, 7.59262829704971978565538093250, 8.316395592242562414120523053257