L(s) = 1 | + (−706. + 159. i)2-s + (−3.25e4 + 1.01e4i)3-s + (4.73e5 − 2.24e5i)4-s − 5.54e6i·5-s + (2.13e7 − 1.23e7i)6-s − 1.49e8i·7-s + (−2.98e8 + 2.34e8i)8-s + (9.56e8 − 6.59e8i)9-s + (8.81e8 + 3.91e9i)10-s − 2.49e6·11-s + (−1.31e10 + 1.21e10i)12-s + 4.88e10·13-s + (2.38e10 + 1.05e11i)14-s + (5.62e10 + 1.80e11i)15-s + (1.73e11 − 2.12e11i)16-s − 4.51e11i·17-s + ⋯ |
L(s) = 1 | + (−0.975 + 0.219i)2-s + (−0.954 + 0.297i)3-s + (0.903 − 0.428i)4-s − 1.26i·5-s + (0.866 − 0.499i)6-s − 1.40i·7-s + (−0.787 + 0.616i)8-s + (0.823 − 0.567i)9-s + (0.278 + 1.23i)10-s − 0.000319·11-s + (−0.735 + 0.677i)12-s + 1.27·13-s + (0.308 + 1.37i)14-s + (0.377 + 1.21i)15-s + (0.632 − 0.774i)16-s − 0.923i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(0.8188439595\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8188439595\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (706. - 159. i)T \) |
| 3 | \( 1 + (3.25e4 - 1.01e4i)T \) |
good | 5 | \( 1 + 5.54e6iT - 1.90e13T^{2} \) |
| 7 | \( 1 + 1.49e8iT - 1.13e16T^{2} \) |
| 11 | \( 1 + 2.49e6T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.88e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 4.51e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 - 6.03e11iT - 1.97e24T^{2} \) |
| 23 | \( 1 - 7.40e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.10e14iT - 6.10e27T^{2} \) |
| 31 | \( 1 + 2.18e14iT - 2.16e28T^{2} \) |
| 37 | \( 1 - 4.99e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.26e13iT - 4.39e30T^{2} \) |
| 43 | \( 1 + 3.26e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 - 1.12e16T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.23e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 6.67e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.62e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 4.88e15iT - 4.95e34T^{2} \) |
| 71 | \( 1 + 2.98e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 9.07e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 3.92e17iT - 1.13e36T^{2} \) |
| 83 | \( 1 + 8.04e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.37e18iT - 1.09e37T^{2} \) |
| 97 | \( 1 + 1.68e18T + 5.60e37T^{2} \) |
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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
−15.67159510847372385567881620191, −13.36369996152024425756583270697, −11.66449846745848513874478141457, −10.48192025496689546080867053793, −9.162766408773730336766296251642, −7.49274975514349247691165122194, −5.89753267853700990660057107889, −4.27043695218947017558480494863, −1.10955991328199985979483585564, −0.52908300604072909689970147585,
1.47575202782955205169706254672, 2.98589023838130873044551780116, 5.91989687619422768008036158263, 6.93023143634435046202128955793, 8.717917449873740446749049421226, 10.56184413750430905943093168568, 11.33007714301454694763587802016, 12.65458710953536999234137131714, 15.06447687743945792509972339687, 16.15256521503625239628646849181