L(s) = 1 | + 3i·3-s + 20.7·5-s + (−11.8 + 14.2i)7-s − 9·9-s − 39.2·11-s − 36.2·13-s + 62.1i·15-s + 92.2i·17-s + 87.5i·19-s + (−42.7 − 35.4i)21-s − 100. i·23-s + 303.·25-s − 27i·27-s + 36.2i·29-s − 264.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.85·5-s + (−0.638 + 0.770i)7-s − 0.333·9-s − 1.07·11-s − 0.773·13-s + 1.06i·15-s + 1.31i·17-s + 1.05i·19-s + (−0.444 − 0.368i)21-s − 0.915i·23-s + 2.42·25-s − 0.192i·27-s + 0.231i·29-s − 1.53·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4947574692\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4947574692\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (11.8 - 14.2i)T \) |
good | 5 | \( 1 - 20.7T + 125T^{2} \) |
| 11 | \( 1 + 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 100. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 36.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 229. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 337. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 185. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 584. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 438. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 722. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 452. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 597. iT - 9.12e5T^{2} \) |
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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
−9.830329721530600655079848964375, −9.052211554208618350471947214727, −8.436003721627422884239493609573, −7.15278857935780041695905824693, −6.03683610313397543084872926144, −5.69096435152608006063636828415, −4.97468690580104181304536944072, −3.56278718136903174955267396110, −2.45231715832679126356415536392, −1.89003336676994963381143372716,
0.098479537623327508470458359295, 1.32979026279787376315420415724, 2.46395862001841282771777235501, 3.04156499253206337454514911095, 4.86103142885190367268934474784, 5.39546192762184169127967521690, 6.35670746486128927254357142704, 7.06982869922909069143668351947, 7.68166039316763454282556236897, 9.073150070242119247522349739347