L(s) = 1 | + (1.58 − 1.21i)2-s + (1.03 − 3.86i)4-s − 1.46·5-s − 6.38i·7-s + (−3.06 − 7.39i)8-s + (−2.32 + 1.78i)10-s + 9.33i·11-s − 22.7·13-s + (−7.77 − 10.1i)14-s + (−13.8 − 8.00i)16-s + 5.17·17-s − 4.35i·19-s + (−1.52 + 5.67i)20-s + (11.3 + 14.8i)22-s − 37.0i·23-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s − 0.293·5-s − 0.912i·7-s + (−0.382 − 0.923i)8-s + (−0.232 + 0.178i)10-s + 0.849i·11-s − 1.74·13-s + (−0.555 − 0.724i)14-s + (−0.866 − 0.500i)16-s + 0.304·17-s − 0.229i·19-s + (−0.0760 + 0.283i)20-s + (0.516 + 0.673i)22-s − 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.284862927\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284862927\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.58 + 1.21i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 1.46T + 25T^{2} \) |
| 7 | \( 1 + 6.38iT - 49T^{2} \) |
| 11 | \( 1 - 9.33iT - 121T^{2} \) |
| 13 | \( 1 + 22.7T + 169T^{2} \) |
| 17 | \( 1 - 5.17T + 289T^{2} \) |
| 23 | \( 1 + 37.0iT - 529T^{2} \) |
| 29 | \( 1 + 23.0T + 841T^{2} \) |
| 31 | \( 1 - 19.9iT - 961T^{2} \) |
| 37 | \( 1 - 24.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 78.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 82.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 86.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 114.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 64.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 25.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 58.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 47.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 51.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 4.37T + 7.92e3T^{2} \) |
| 97 | \( 1 - 16.1T + 9.40e3T^{2} \) |
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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
−9.957792842893130731226515384488, −9.395012370914132609769575815258, −7.75773355946988513852273162727, −7.15543288464858484163576257235, −6.14046129539264637845072281595, −4.73463448839271896231329205595, −4.43189927135947574742011344785, −3.08909253546949800048266615692, −1.96161202540565155615521471194, −0.32182551158298891553853124470,
2.26654930217766025634622303377, 3.28974722474987458269701343126, 4.40618874819247040160200044675, 5.56482027131936608490034884917, 5.92924425133276875151372631901, 7.46067644378494103371680120409, 7.71486864305507290139426447563, 8.953643589773579220471039313061, 9.651873167140332935153430626932, 11.07023117588822653505045944483