L(s) = 1 | − 408. i·3-s + 5.36e3·5-s − 9.54e3i·7-s − 1.07e5·9-s − 1.01e5i·11-s + 3.74e4·13-s − 2.19e6i·15-s − 1.39e6·17-s + 2.17e6i·19-s − 3.89e6·21-s − 4.54e6i·23-s + 1.90e7·25-s + 1.98e7i·27-s + 7.75e6·29-s + 4.37e7i·31-s + ⋯ |
L(s) = 1 | − 1.68i·3-s + 1.71·5-s − 0.567i·7-s − 1.82·9-s − 0.629i·11-s + 0.100·13-s − 2.88i·15-s − 0.985·17-s + 0.877i·19-s − 0.954·21-s − 0.705i·23-s + 1.94·25-s + 1.38i·27-s + 0.378·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.04620 - 1.81207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04620 - 1.81207i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 408. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 5.36e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 9.54e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.01e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 3.74e4T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.39e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 2.17e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 4.54e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 7.75e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 4.37e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 2.45e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.44e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.85e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 5.16e6iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 2.90e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 7.53e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 5.34e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 8.34e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.40e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.64e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 2.71e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 4.09e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 4.62e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 2.28e9T + 7.37e19T^{2} \) |
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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
−16.78812280568250263333322473177, −14.10681470129994491281575411642, −13.58330032109512062695845732692, −12.48131386074017299369512442675, −10.58286418060353812232858296586, −8.700346315919732513757505959186, −6.89964890220859404897507991748, −5.84958137220950197859296000347, −2.31777551149037361156197018100, −1.04463879815843202073031552507,
2.45967805455503693161608709367, 4.68758227200634109429992512006, 5.98059704988160615002328850388, 9.147324399292433709100372209070, 9.755150154559795966936835243180, 11.07874599739570773075978228905, 13.27691570971726492594134170083, 14.67418699120793804235137948633, 15.70680270759145709406398360734, 17.08754408970829694858334836338