L(s) = 1 | + 3.42·5-s − 0.505i·7-s + 3.31i·11-s + 5.43i·13-s − 1.58i·17-s + 6.74·19-s − 4.30·23-s + 6.74·25-s − 2.55·29-s + 4.92i·31-s − 1.73i·35-s − 7.45i·37-s − 8.51i·41-s + 4·43-s − 5.10·47-s + ⋯ |
L(s) = 1 | + 1.53·5-s − 0.191i·7-s + 1.00i·11-s + 1.50i·13-s − 0.384i·17-s + 1.54·19-s − 0.897·23-s + 1.34·25-s − 0.473·29-s + 0.884i·31-s − 0.292i·35-s − 1.22i·37-s − 1.32i·41-s + 0.609·43-s − 0.744·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00111 + 0.476960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00111 + 0.476960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 + 0.505iT - 7T^{2} \) |
| 11 | \( 1 - 3.31iT - 11T^{2} \) |
| 13 | \( 1 - 5.43iT - 13T^{2} \) |
| 17 | \( 1 + 1.58iT - 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 4.92iT - 31T^{2} \) |
| 37 | \( 1 + 7.45iT - 37T^{2} \) |
| 41 | \( 1 + 8.51iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.10T + 47T^{2} \) |
| 53 | \( 1 + 0.875T + 53T^{2} \) |
| 59 | \( 1 + 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 + 6.44iT - 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.944331275496705268624811886623, −9.533642585026603059696052687436, −8.878224516542773732710747223726, −7.40974085387741903247650642554, −6.84979388065812458844720203631, −5.80835986202079524678315891563, −5.05250968492104765993016654074, −3.93412429774273470551505652582, −2.39032931511916959826561136391, −1.58058285366032570293682498013,
1.13770639634418469473904801268, 2.53210870945889866503666680983, 3.45022242313395660766719518460, 5.12071424278952179788837346623, 5.77285938986329475145816085425, 6.30192647023618178390957114805, 7.69966161101180956875952283829, 8.406319227887731630066465571644, 9.523463729033639652814765695432, 9.933809297147362298332415074661