| L(s) = 1 | + (50.0 − 86.6i)3-s + (−5.80e3 − 1.00e4i)5-s + (−3.70e4 − 2.46e4i)7-s + (8.35e4 + 1.44e5i)9-s + (−7.70e4 + 1.33e5i)11-s + 3.07e5·13-s − 1.16e6·15-s + (−1.27e6 + 2.20e6i)17-s + (−1.01e6 − 1.74e6i)19-s + (−3.98e6 + 1.97e6i)21-s + (2.18e7 + 3.78e7i)23-s + (−4.30e7 + 7.45e7i)25-s + 3.44e7·27-s − 1.95e8·29-s + (−3.20e7 + 5.54e7i)31-s + ⋯ |
| L(s) = 1 | + (0.118 − 0.205i)3-s + (−0.831 − 1.43i)5-s + (−0.832 − 0.553i)7-s + (0.471 + 0.817i)9-s + (−0.144 + 0.249i)11-s + 0.229·13-s − 0.395·15-s + (−0.217 + 0.376i)17-s + (−0.0936 − 0.162i)19-s + (−0.212 + 0.105i)21-s + (0.707 + 1.22i)23-s + (−0.881 + 1.52i)25-s + 0.461·27-s − 1.76·29-s + (−0.200 + 0.348i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.133189 + 0.200318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133189 + 0.200318i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (3.70e4 + 2.46e4i)T \) |
| good | 3 | \( 1 + (-50.0 + 86.6i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (5.80e3 + 1.00e4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (7.70e4 - 1.33e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 3.07e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (1.27e6 - 2.20e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.01e6 + 1.74e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.18e7 - 3.78e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.95e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (3.20e7 - 5.54e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.79e8 - 4.84e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 7.18e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.29e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (6.23e8 + 1.07e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.53e9 + 2.66e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.70e9 + 6.41e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.38e9 + 4.13e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (4.18e9 - 7.25e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.46e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.63e10 - 2.84e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (8.02e9 + 1.38e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 6.64e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.26e10 - 7.39e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 5.10e10T + 7.15e21T^{2} \) |
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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
−15.34341424933271002535822597610, −13.26093660750147235341777294699, −12.92006128319644425442726486495, −11.38342675065678025536207848413, −9.741967538221127314706873302838, −8.364400985797787373424693475963, −7.19423297450477980815936587613, −5.09358067044267503928130395286, −3.77089513620585057242822713781, −1.41414861835056284921400625881,
0.086462847157811971951535352565, 2.78784347781601193749581971938, 3.84154581633505453958428687355, 6.23824289814088050746246211300, 7.30810852017670950417158471062, 9.097908034427312108187391597599, 10.46466761525430633502210069403, 11.63105749218043824760926942987, 12.95030402301566017533644209187, 14.69052698416160839920161713805
