L(s) = 1 | + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)7-s + (2.5 − 4.33i)11-s + (−2 − 3.46i)13-s + 8·17-s + 2·19-s + (1 + 1.73i)23-s + (2 − 3.46i)25-s + (3 − 5.19i)29-s + (3.5 + 6.06i)31-s + 3·35-s − 6·37-s + (−3 − 5.19i)41-s + (1 − 1.73i)43-s + (3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.753 − 1.30i)11-s + (−0.554 − 0.960i)13-s + 1.94·17-s + 0.458·19-s + (0.208 + 0.361i)23-s + (0.400 − 0.692i)25-s + (0.557 − 0.964i)29-s + (0.628 + 1.08i)31-s + 0.507·35-s − 0.986·37-s + (−0.468 − 0.811i)41-s + (0.152 − 0.264i)43-s + (0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30546 - 0.475150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30546 - 0.475150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.29425797916683476209565869922, −9.626758317022921323053723703404, −8.593886958864914171241138517523, −8.110546386813897342502550147185, −6.84029613150383195588173883153, −5.74709869924776261562529017044, −5.24767772441700568968181432170, −3.59906237316680168958216947870, −2.85537760843342155178755531408, −0.888267160271876403026464002014,
1.37186484275225425611946086557, 3.07476193194455228130293936967, 4.04682375699432952968016193007, 5.00692128552588531847269946150, 6.46245180557132342142837175121, 7.14958929160831739874800294386, 7.70722650355393378263148223792, 9.179920002163070702439689215692, 9.866151249978199832985838671133, 10.42583645999444522620363738423