L(s) = 1 | − 31.0i·2-s + 76.9i·3-s − 710.·4-s − 309. i·5-s + 2.39e3·6-s + 1.60e3i·7-s + 1.41e4i·8-s + 638.·9-s − 9.61e3·10-s − 7.35e3·11-s − 5.47e4i·12-s + 4.21e4·13-s + 4.98e4·14-s + 2.37e4·15-s + 2.57e5·16-s − 5.70e4·17-s + ⋯ |
L(s) = 1 | − 1.94i·2-s + 0.950i·3-s − 2.77·4-s − 0.494i·5-s + 1.84·6-s + 0.668i·7-s + 3.45i·8-s + 0.0972·9-s − 0.961·10-s − 0.502·11-s − 2.63i·12-s + 1.47·13-s + 1.29·14-s + 0.469·15-s + 3.93·16-s − 0.683·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.26735 - 0.848317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26735 - 0.848317i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-1.30e6 - 3.16e6i)T \) |
good | 2 | \( 1 + 31.0iT - 256T^{2} \) |
| 3 | \( 1 - 76.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 309. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.60e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 7.35e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 4.21e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 5.70e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.27e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.87e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 2.92e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 4.78e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.22e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 5.52e6T + 7.98e12T^{2} \) |
| 47 | \( 1 + 1.51e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 7.83e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.53e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 3.41e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 3.26e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.27e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.32e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.22e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.72e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 7.79e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.35e7T + 7.83e15T^{2} \) |
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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
−13.36824650252867738291705700179, −12.73076830729056503640614780788, −11.24921701291759686572748862708, −10.65889788801733990916489414497, −9.250894056925344543320244518652, −8.738066886685802819212988304882, −5.21087371373387757136099802142, −4.17227401587825534740563461316, −2.79416524137440292235255615825, −1.06562537123398876859868930986,
0.813803040489193088465429346494, 4.04564924177263132108824199850, 5.85786490846415572029457546668, 6.88889230928686897369316275236, 7.65050576563317165086329279330, 8.839487118170347629867327935261, 10.55883895699017714486874631367, 12.88575540798253069399539766778, 13.49887262655069198005964061833, 14.47814159148047133146675356872