| L(s) = 1 | + (−121.5 + 210. i)3-s + (−6.71e3 − 1.16e4i)5-s + (4.14e4 − 1.61e4i)7-s + (−2.95e4 − 5.11e4i)9-s + (−4.25e5 + 7.37e5i)11-s + 2.13e6·13-s + 3.26e6·15-s + (1.28e6 − 2.21e6i)17-s + (8.87e6 + 1.53e7i)19-s + (−1.64e6 + 1.06e7i)21-s + (−1.81e7 − 3.14e7i)23-s + (−6.58e7 + 1.14e8i)25-s + 1.43e7·27-s + 5.71e5·29-s + (1.07e8 − 1.86e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.961 − 1.66i)5-s + (0.932 − 0.362i)7-s + (−0.166 − 0.288i)9-s + (−0.796 + 1.38i)11-s + 1.59·13-s + 1.11·15-s + (0.218 − 0.379i)17-s + (0.822 + 1.42i)19-s + (−0.0878 + 0.570i)21-s + (−0.588 − 1.01i)23-s + (−1.34 + 2.33i)25-s + 0.192·27-s + 0.00517·29-s + (0.676 − 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.26634 - 0.926602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26634 - 0.926602i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (121.5 - 210. i)T \) |
| 7 | \( 1 + (-4.14e4 + 1.61e4i)T \) |
| good | 5 | \( 1 + (6.71e3 + 1.16e4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (4.25e5 - 7.37e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 2.13e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.28e6 + 2.21e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.87e6 - 1.53e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.81e7 + 3.14e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 5.71e5T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.07e8 + 1.86e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (9.87e7 + 1.71e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 2.72e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.11e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-3.98e8 - 6.90e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (6.76e8 - 1.17e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.81e8 + 6.60e8i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.04e9 + 3.53e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.08e9 + 1.88e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.82e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.03e10 + 1.78e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.28e10 + 2.22e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 1.36e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-5.18e9 - 8.98e9i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.14e11T + 7.15e21T^{2} \) |
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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
−11.92164435953700314297447318936, −10.81763159404177354195398350334, −9.564504867221673987805762009931, −8.287853519324363863669304865097, −7.72366628926834167168630386553, −5.61085618492136178075296161444, −4.59686486486977173940751129994, −3.94131043265854109786533319695, −1.57484908991724084291164796642, −0.52712604864921260805944883943,
0.967173764543879807671148576321, 2.69276859607702813567663052807, 3.63728505483400114346695428056, 5.51292443099700592328236816725, 6.60970335771930666820950313553, 7.76007213198276241918085642738, 8.488129726175601106895141915193, 10.66013418000173460963610558227, 11.17356330789647370067149913022, 11.79356726426309925662606223795
