| L(s) = 1 | + (−121.5 − 210. i)3-s + (−5.53e3 + 9.59e3i)5-s + (−3.20e4 − 3.08e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (4.51e5 + 7.82e5i)11-s − 7.61e5·13-s + 2.69e6·15-s + (2.39e6 + 4.14e6i)17-s + (−2.25e6 + 3.90e6i)19-s + (−2.60e6 + 1.04e7i)21-s + (−2.38e7 + 4.13e7i)23-s + (−3.69e7 − 6.40e7i)25-s + 1.43e7·27-s + 1.65e8·29-s + (−2.95e7 − 5.11e7i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.792 + 1.37i)5-s + (−0.720 − 0.693i)7-s + (−0.166 + 0.288i)9-s + (0.845 + 1.46i)11-s − 0.568·13-s + 0.915·15-s + (0.408 + 0.707i)17-s + (−0.208 + 0.361i)19-s + (−0.139 + 0.560i)21-s + (−0.773 + 1.34i)23-s + (−0.757 − 1.31i)25-s + 0.192·27-s + 1.49·29-s + (−0.185 − 0.321i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0390225 - 0.0842711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0390225 - 0.0842711i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (3.20e4 + 3.08e4i)T \) |
| good | 5 | \( 1 + (5.53e3 - 9.59e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.51e5 - 7.82e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.61e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.39e6 - 4.14e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.25e6 - 3.90e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.38e7 - 4.13e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.65e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (2.95e7 + 5.11e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.08e8 + 1.87e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.52e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.04e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (6.93e7 - 1.20e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.34e9 - 4.05e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (4.49e9 + 7.78e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (1.08e8 - 1.87e8i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (8.64e9 + 1.49e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 5.28e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (5.55e9 + 9.61e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-2.14e9 + 3.71e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.61e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (8.53e9 - 1.47e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.57e11T + 7.15e21T^{2} \) |
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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
−12.44154615634166519508798650361, −11.78025907209263415315664793238, −10.50945501019021903601445158615, −9.768554607065371128601772486398, −7.76813511680059574706158630837, −7.08027414773079277068989184285, −6.27926498650771006932834797517, −4.26941234289570533492608424557, −3.22544646253221817548704109614, −1.71079973057188848559060575103,
0.03120609512125453469326498569, 0.872300064039292173025284101546, 3.00957096044531033836221040057, 4.26582934555001369584350333958, 5.32177364212949523524443919114, 6.52857676431261970857254628821, 8.413014284073783252813584337965, 8.896812247846353668563122871487, 10.12084767475639802746015273936, 11.77537799850268108589636053366
