L(s) = 1 | + (−25.8 + 18.8i)2-s + (−88.4 − 272. i)3-s + (316. − 973. i)4-s + (−354. − 257. i)5-s + (7.40e3 + 5.38e3i)6-s + (1.23e4 − 3.78e4i)7-s + (1.01e4 + 3.11e4i)8-s + (7.70e4 − 5.60e4i)9-s + 1.40e4·10-s + (7.66e4 + 5.28e5i)11-s − 2.92e5·12-s + (−3.03e5 + 2.20e5i)13-s + (3.93e5 + 1.21e6i)14-s + (−3.87e4 + 1.19e5i)15-s + (−8.48e5 − 6.16e5i)16-s + (−4.83e6 − 3.51e6i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.210 − 0.646i)3-s + (0.154 − 0.475i)4-s + (−0.0507 − 0.0368i)5-s + (0.388 + 0.282i)6-s + (0.276 − 0.851i)7-s + (0.109 + 0.336i)8-s + (0.435 − 0.316i)9-s + 0.0443·10-s + (0.143 + 0.989i)11-s − 0.339·12-s + (−0.226 + 0.164i)13-s + (0.195 + 0.602i)14-s + (−0.0131 + 0.0405i)15-s + (−0.202 − 0.146i)16-s + (−0.826 − 0.600i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.121366 - 0.549878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.121366 - 0.549878i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (25.8 - 18.8i)T \) |
| 11 | \( 1 + (-7.66e4 - 5.28e5i)T \) |
good | 3 | \( 1 + (88.4 + 272. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (354. + 257. i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-1.23e4 + 3.78e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (3.03e5 - 2.20e5i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (4.83e6 + 3.51e6i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (3.86e6 + 1.19e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 + 2.09e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (3.69e7 - 1.13e8i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (1.81e8 - 1.32e8i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (6.87e6 - 2.11e7i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-2.52e7 - 7.77e7i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 - 2.17e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (2.30e8 + 7.09e8i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (1.53e9 - 1.11e9i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-2.00e8 + 6.15e8i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-2.11e8 - 1.53e8i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 + 1.86e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (1.66e10 + 1.20e10i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (-6.06e9 + 1.86e10i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-2.85e10 + 2.07e10i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (-3.73e10 - 2.71e10i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 - 4.74e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (8.06e10 - 5.85e10i)T + (2.21e21 - 6.80e21i)T^{2} \) |
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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
−14.95549690508104010978484897901, −13.56803414160202626721373037775, −12.19410815796030489726712932270, −10.66858577352753875284940610944, −9.249356484642107283084990390971, −7.46687266724657476720820494582, −6.70114247260302708118050851431, −4.53913480784054345639147622463, −1.77343080661188453513821059238, −0.26138745581047077450542518630,
1.94632675360472848836926019198, 3.90513344624750082369322285049, 5.78353102295504063995645884232, 7.958096544745408303369716344894, 9.277701379597682166103996969384, 10.59836530198411637804011590279, 11.65027783093525078678701812334, 13.12925373662950451716289903283, 14.95302494753206361253307527509, 16.06088527004298607822656116234