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
−9.485813370399958563919502967031, −8.951634556234897572148547207002, −8.518257978915396973390207937283, −7.63361472748813398747316296847, −6.44331345847224617516126821105, −5.53728175851661979964076402498, −4.43344731442503155807749928547, −3.68059514086585956584232506797, −2.75294989911971193326437838504, −1.21937403827879994524445922281,
1.11716901493441856929418489468, 2.51827260452853470544261815644, 3.57019441339554882144124671223, 4.07179283146753526843208788757, 5.67908550775228635526184470320, 6.74979576627629701340841648961, 7.43857075431756871111942271447, 7.74599525251996953835690431307, 8.978420142779987815316409079964, 9.799444524831515317445240374411