L(s) = 1 | + (−6.87 − 44.7i)2-s + (−291. + 291. i)3-s + (−1.95e3 + 614. i)4-s + (−9.75e3 − 9.75e3i)5-s + (1.50e4 + 1.10e4i)6-s − 1.64e4i·7-s + (4.09e4 + 8.31e4i)8-s + 7.59e3i·9-s + (−3.69e5 + 5.03e5i)10-s + (4.77e5 + 4.77e5i)11-s + (3.89e5 − 7.47e5i)12-s + (6.65e5 − 6.65e5i)13-s + (−7.35e5 + 1.13e5i)14-s + 5.68e6·15-s + (3.43e6 − 2.40e6i)16-s − 5.80e6·17-s + ⋯ |
L(s) = 1 | + (−0.151 − 0.988i)2-s + (−0.691 + 0.691i)3-s + (−0.953 + 0.300i)4-s + (−1.39 − 1.39i)5-s + (0.788 + 0.578i)6-s − 0.369i·7-s + (0.441 + 0.897i)8-s + 0.0428i·9-s + (−1.16 + 1.59i)10-s + (0.894 + 0.894i)11-s + (0.452 − 0.867i)12-s + (0.497 − 0.497i)13-s + (−0.365 + 0.0561i)14-s + 1.93·15-s + (0.819 − 0.572i)16-s − 0.991·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.591086 + 0.0644256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.591086 + 0.0644256i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (6.87 + 44.7i)T \) |
good | 3 | \( 1 + (291. - 291. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (9.75e3 + 9.75e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 + 1.64e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-4.77e5 - 4.77e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (-6.65e5 + 6.65e5i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 + 5.80e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-1.91e6 + 1.91e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 1.46e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.83e7 + 5.83e7i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 - 8.50e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-2.52e7 - 2.52e7i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 9.24e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.00e9 - 1.00e9i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 2.86e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-2.34e9 - 2.34e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (5.55e8 + 5.55e8i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (2.31e9 - 2.31e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-1.02e10 + 1.02e10i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.39e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 2.22e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.84e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (4.10e9 - 4.10e9i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 1.47e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 9.56e10T + 7.15e21T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.61155345688729975558247129858, −15.45849866108879202889339067957, −13.19220433221970275822488914824, −11.96437200982129139274051430067, −11.10987030010882185761883319020, −9.467490812243759060843905066770, −8.051858971059591550111260742865, −4.78799471396137892998727363052, −4.05992341204255417957734698531, −0.995934483925858543306471467300,
0.41382110665762507734977227550, 3.80308871922283397934015659951, 6.30378920264518332829792443796, 7.01062721938748318398556172146, 8.598577710935893186635088828787, 10.98223089738072294385551404440, 12.06393291096726491026599303305, 14.02655667455599218732341706899, 15.17421601112462452383760919539, 16.25052344069052273767318713261