L(s) = 1 | + (−3.20e4 + 5.71e4i)2-s + (−2.24e9 − 3.66e9i)4-s + 4.45e10·5-s − 4.42e13i·7-s + (2.81e14 − 1.12e13i)8-s + (−1.42e15 + 2.54e15i)10-s − 2.28e16i·11-s − 7.09e17·13-s + (2.53e18 + 1.41e18i)14-s + (−8.36e18 + 1.64e19i)16-s + 5.13e19·17-s + 9.08e19i·19-s + (−1.00e20 − 1.63e20i)20-s + (1.30e21 + 7.32e20i)22-s − 3.40e21i·23-s + ⋯ |
L(s) = 1 | + (−0.488 + 0.872i)2-s + (−0.522 − 0.852i)4-s + 0.292·5-s − 1.33i·7-s + (0.999 − 0.0398i)8-s + (−0.142 + 0.254i)10-s − 0.497i·11-s − 1.06·13-s + (1.16 + 0.650i)14-s + (−0.453 + 0.891i)16-s + 1.05·17-s + 0.315i·19-s + (−0.152 − 0.249i)20-s + (0.434 + 0.243i)22-s − 0.555i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.5342288161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5342288161\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.20e4 - 5.71e4i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.45e10T + 2.32e22T^{2} \) |
| 7 | \( 1 + 4.42e13iT - 1.10e27T^{2} \) |
| 11 | \( 1 + 2.28e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 + 7.09e17T + 4.42e35T^{2} \) |
| 17 | \( 1 - 5.13e19T + 2.36e39T^{2} \) |
| 19 | \( 1 - 9.08e19iT - 8.31e40T^{2} \) |
| 23 | \( 1 + 3.40e21iT - 3.76e43T^{2} \) |
| 29 | \( 1 - 9.75e22T + 6.26e46T^{2} \) |
| 31 | \( 1 - 8.90e23iT - 5.29e47T^{2} \) |
| 37 | \( 1 + 1.36e25T + 1.52e50T^{2} \) |
| 41 | \( 1 + 9.45e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + 2.05e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 - 6.70e25iT - 3.21e53T^{2} \) |
| 53 | \( 1 - 3.12e27T + 1.50e55T^{2} \) |
| 59 | \( 1 - 4.03e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 4.61e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + 2.83e29iT - 2.71e58T^{2} \) |
| 71 | \( 1 + 3.78e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + 8.28e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 3.22e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + 3.96e29iT - 2.57e61T^{2} \) |
| 89 | \( 1 + 1.44e31T + 2.40e62T^{2} \) |
| 97 | \( 1 - 5.69e30T + 3.77e63T^{2} \) |
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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
−10.44530603891595672789696306363, −10.09247566473069438267840654134, −8.721369257217280494157508541386, −7.59831816124542536261542335978, −6.89743099887977459670555331796, −5.66764847668914506351738192690, −4.65999279062966783133540942493, −3.45019202114981505410669800370, −1.73102713860314203366565113120, −0.73421110655071246624286359290,
0.14350554819315498840121461918, 1.59161657041963512820517904123, 2.33853372742836858429389971553, 3.25819229116480846763569650729, 4.70538416544558397358921218091, 5.69462149815896543291574482335, 7.31666866475221409634163435839, 8.376268133820036394011002258755, 9.507808609388374272874966959458, 10.04269534707762972708170308432