L(s) = 1 | + (1.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1.5 − 2.59i)9-s + (−2.5 − 4.33i)11-s − 1.73i·15-s + 3·17-s − 5·19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s − 5.19i·27-s + (5 + 8.66i)29-s + (−1 + 1.73i)31-s + (−7.5 − 4.33i)33-s + 4·37-s + (1.5 − 2.59i)41-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.5 − 0.866i)9-s + (−0.753 − 1.30i)11-s − 0.447i·15-s + 0.727·17-s − 1.14·19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.999i·27-s + (0.928 + 1.60i)29-s + (−0.179 + 0.311i)31-s + (−1.30 − 0.753i)33-s + 0.657·37-s + (0.234 − 0.405i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52251 - 1.27753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52251 - 1.27753i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 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
−10.26020690497743039708356183530, −9.012308300768752647510164394594, −8.575926059178326469351972147858, −7.82352083783939644058943236720, −6.74371997742015913801222992217, −5.86297783752437163521518093388, −4.69898396479728600753142460449, −3.39268277321198310993092640098, −2.50226890156502896041659072926, −0.977723197946596028285680891564,
1.98991972099260763786963171994, 2.89651450367539122506627930403, 4.13254258241428061702997466158, 4.97085517556857291745178790127, 6.18736440487051839131936384388, 7.44372152204783509840973182660, 7.87465109184261690182734328560, 9.025303657793254598679494721108, 9.841654842723544589284375624556, 10.30167182590907834215945763363