L(s) = 1 | + (1.5 − 0.866i)3-s + (−1 + 1.73i)5-s + (−1 − 1.73i)7-s + (1.5 − 2.59i)9-s + (2.5 + 4.33i)11-s + (2 − 3.46i)13-s + 3.46i·15-s + 17-s + 5·19-s + (−3 − 1.73i)21-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 5.19i·27-s + (−3 − 5.19i)29-s + (7.5 + 4.33i)33-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.377 − 0.654i)7-s + (0.5 − 0.866i)9-s + (0.753 + 1.30i)11-s + (0.554 − 0.960i)13-s + 0.894i·15-s + 0.242·17-s + 1.14·19-s + (−0.654 − 0.377i)21-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 0.999i·27-s + (−0.557 − 0.964i)29-s + (1.30 + 0.753i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.168388835\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.168388835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 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 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.799444524831515317445240374411, −8.978420142779987815316409079964, −7.74599525251996953835690431307, −7.43857075431756871111942271447, −6.74979576627629701340841648961, −5.67908550775228635526184470320, −4.07179283146753526843208788757, −3.57019441339554882144124671223, −2.51827260452853470544261815644, −1.11716901493441856929418489468,
1.21937403827879994524445922281, 2.75294989911971193326437838504, 3.68059514086585956584232506797, 4.43344731442503155807749928547, 5.53728175851661979964076402498, 6.44331345847224617516126821105, 7.63361472748813398747316296847, 8.518257978915396973390207937283, 8.951634556234897572148547207002, 9.485813370399958563919502967031