L(s) = 1 | + (1.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)7-s + (1.5 − 2.59i)9-s + (2 − 3.46i)11-s + (2 + 3.46i)13-s + (−1.5 − 0.866i)15-s + 2·17-s + 2·19-s + 5.19i·21-s + (3.5 + 6.06i)23-s + (−0.499 + 0.866i)25-s − 5.19i·27-s + (4.5 − 7.79i)29-s + (−3 − 5.19i)31-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.5 − 0.866i)9-s + (0.603 − 1.04i)11-s + (0.554 + 0.960i)13-s + (−0.387 − 0.223i)15-s + 0.485·17-s + 0.458·19-s + 1.13i·21-s + (0.729 + 1.26i)23-s + (−0.0999 + 0.173i)25-s − 0.999i·27-s + (0.835 − 1.44i)29-s + (−0.538 − 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.277879680\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.277879680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.5 - 6.06i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.286516392215275375626881020447, −8.684205340604873195326364701413, −8.034347358198727872612474359148, −7.03247170236136401176869984337, −6.20910172218835029345130070436, −5.48529482331468642891240993658, −3.99332336757437231258778983069, −3.36095979141425679052665092176, −2.27032257874278009754569112778, −1.02074679397857580838969339398,
1.24681582128021905837638373273, 2.84595659644318422784676185891, 3.49435726826145050355321227370, 4.34169950456451886517474604303, 5.25545604249761292761246106319, 6.77301527900561703522936967243, 7.09647897740824745320782206073, 8.077001323888815233106077750829, 8.820509884844381138709402753007, 9.711676928177112821308923450659