L(s) = 1 | + (−257. + 254. i)2-s + 4.83e3i·3-s + (1.83e3 − 1.31e5i)4-s + 5.24e5i·5-s + (−1.22e6 − 1.24e6i)6-s + 1.57e7·7-s + (3.28e7 + 3.42e7i)8-s + 1.05e8·9-s + (−1.33e8 − 1.35e8i)10-s − 2.44e8i·11-s + (6.33e8 + 8.89e6i)12-s + 2.66e9i·13-s + (−4.06e9 + 4.00e9i)14-s − 2.53e9·15-s + (−1.71e10 − 4.82e8i)16-s − 2.82e10·17-s + ⋯ |
L(s) = 1 | + (−0.712 + 0.702i)2-s + 0.425i·3-s + (0.0140 − 0.999i)4-s + 0.600i·5-s + (−0.298 − 0.302i)6-s + 1.03·7-s + (0.692 + 0.721i)8-s + 0.819·9-s + (−0.421 − 0.427i)10-s − 0.344i·11-s + (0.425 + 0.00597i)12-s + 0.905i·13-s + (−0.735 + 0.725i)14-s − 0.255·15-s + (−0.999 − 0.0280i)16-s − 0.983·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(0.509259 + 1.19376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.509259 + 1.19376i\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (257. - 254. i)T \) |
good | 3 | \( 1 - 4.83e3iT - 1.29e8T^{2} \) |
| 5 | \( 1 - 5.24e5iT - 7.62e11T^{2} \) |
| 7 | \( 1 - 1.57e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 2.44e8iT - 5.05e17T^{2} \) |
| 13 | \( 1 - 2.66e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + 2.82e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.79e10iT - 5.48e21T^{2} \) |
| 23 | \( 1 + 3.66e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.78e11iT - 7.25e24T^{2} \) |
| 31 | \( 1 - 3.68e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.50e13iT - 4.56e26T^{2} \) |
| 41 | \( 1 + 3.67e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.25e14iT - 5.87e27T^{2} \) |
| 47 | \( 1 - 1.02e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 4.67e14iT - 2.05e29T^{2} \) |
| 59 | \( 1 + 4.08e13iT - 1.27e30T^{2} \) |
| 61 | \( 1 + 2.55e15iT - 2.24e30T^{2} \) |
| 67 | \( 1 - 2.46e15iT - 1.10e31T^{2} \) |
| 71 | \( 1 - 1.09e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.21e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.49e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.68e16iT - 4.21e32T^{2} \) |
| 89 | \( 1 + 4.15e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 7.13e15T + 5.95e33T^{2} \) |
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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
−17.98190683882842427828968799267, −16.47716530430271911043348914611, −15.17837605235193254000600980587, −14.03959583915538396705101094963, −11.26525425847194885656783100357, −9.950519099825975659597453863265, −8.224533239440868793513630980173, −6.60658437872296856327630208852, −4.57925033453795506103620844530, −1.63999979014838998478265782047,
0.71467164609970527284521361178, 2.06939227516634202756925143554, 4.50754750346270463133513197676, 7.39614835167933185159193148691, 8.770471221371494046032967043329, 10.56361089229579259669684455571, 12.13282122580669036074339836797, 13.32873622387385118068800007655, 15.65222640647288739353614054445, 17.44210843016758760201825051990