| L(s) = 1 | + (−24.4 − 42.2i)2-s + (129. − 400. i)3-s + (−168. + 291. i)4-s + (5.96e3 − 1.03e4i)5-s + (−2.01e4 + 4.29e3i)6-s + (1.95e4 + 3.38e4i)7-s − 8.35e4·8-s + (−1.43e5 − 1.03e5i)9-s − 5.82e5·10-s + (1.93e5 + 3.35e5i)11-s + (9.50e4 + 1.05e5i)12-s + (−8.43e4 + 1.46e5i)13-s + (9.53e5 − 1.65e6i)14-s + (−3.36e6 − 3.72e6i)15-s + (2.38e6 + 4.13e6i)16-s + 1.08e7·17-s + ⋯ |
| L(s) = 1 | + (−0.539 − 0.934i)2-s + (0.307 − 0.951i)3-s + (−0.0823 + 0.142i)4-s + (0.853 − 1.47i)5-s + (−1.05 + 0.225i)6-s + (0.439 + 0.760i)7-s − 0.901·8-s + (−0.810 − 0.585i)9-s − 1.84·10-s + (0.362 + 0.627i)11-s + (0.110 + 0.122i)12-s + (−0.0629 + 0.109i)13-s + (0.474 − 0.821i)14-s + (−1.14 − 1.26i)15-s + (0.568 + 0.985i)16-s + 1.84·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0723i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0555323 - 1.53304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0555323 - 1.53304i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-129. + 400. i)T \) |
| good | 2 | \( 1 + (24.4 + 42.2i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-5.96e3 + 1.03e4i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-1.95e4 - 3.38e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.93e5 - 3.35e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (8.43e4 - 1.46e5i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 - 1.08e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.47e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-2.90e6 + 5.02e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (7.85e6 + 1.35e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (5.53e6 - 9.59e6i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 - 3.96e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-3.81e8 + 6.60e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-1.72e8 - 2.98e8i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (7.83e8 + 1.35e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + 2.93e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (1.02e8 - 1.77e8i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.00e9 - 5.20e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (5.16e9 - 8.94e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.82e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.48e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-2.27e10 - 3.93e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (4.70e9 + 8.14e9i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 + 8.01e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (9.77e9 + 1.69e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−18.15552117394255799619371239583, −17.02233470835462343455244089684, −14.60905040584368237322009543973, −12.74316351540218384294812764868, −11.97963137502705373925711306778, −9.615326916449532914330940079483, −8.495244278843631501207817239751, −5.71700027383789824218480612325, −2.10577949231244977054455072219, −1.05426016837773823624344117039,
3.17601299366776372845003064972, 6.06809972527351959674562530306, 7.78616839165745073503341911112, 9.699322481951087637645061904173, 10.97164384827835146471484441612, 14.18973914832876863608721369620, 14.83552319082470343923631990059, 16.53254150047907899485711945012, 17.47599806595112005908197017943, 18.92578901040103090807284671834
