L(s) = 1 | + (−0.5 − 0.866i)5-s + (−2.5 + 4.33i)11-s − 3·17-s + 5·19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + (−5 + 8.66i)29-s + (1 + 1.73i)31-s + 4·37-s + (−1.5 − 2.59i)41-s + (−1.5 + 2.59i)43-s + (2 − 3.46i)47-s + (3.5 + 6.06i)49-s + 6·53-s + 5·55-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.753 + 1.30i)11-s − 0.727·17-s + 1.14·19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + (−0.928 + 1.60i)29-s + (0.179 + 0.311i)31-s + 0.657·37-s + (−0.234 − 0.405i)41-s + (−0.228 + 0.396i)43-s + (0.291 − 0.505i)47-s + (0.5 + 0.866i)49-s + 0.824·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147481398\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147481398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 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
−9.923563652643577875310518289806, −9.315041163704273039579873028536, −8.448724820006663224210405519239, −7.36291071449708716212659163074, −7.07016566737228500032490693960, −5.54365566509771538340622420165, −4.98099160613752284386215235684, −3.94209518255421705959660601965, −2.74230156776671143086075635522, −1.44122129871668389146920094671,
0.52786423627989410920559634578, 2.41506223502220968744004461809, 3.29675854143894482553301155332, 4.39603607525990205970093455395, 5.51186283950166454204954400442, 6.25184013947492097002411658442, 7.26834966634586408410464585410, 8.049922546873995106559747221301, 8.788706251432360872705215774460, 9.737554885136418086752983568606