| L(s) = 1 | + (3 + 4.24i)3-s + 5.65i·5-s + 16.9i·7-s + (−8.99 + 25.4i)9-s + 30·11-s + 72·13-s + (−24 + 16.9i)15-s + 50.9i·17-s + 25.4i·19-s + (−71.9 + 50.9i)21-s + 144·23-s + 93·25-s + (−134. + 38.1i)27-s − 5.65i·29-s − 220. i·31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s + 0.505i·5-s + 0.916i·7-s + (−0.333 + 0.942i)9-s + 0.822·11-s + 1.53·13-s + (−0.413 + 0.292i)15-s + 0.726i·17-s + 0.307i·19-s + (−0.748 + 0.529i)21-s + 1.30·23-s + 0.743·25-s + (−0.962 + 0.272i)27-s − 0.0362i·29-s − 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.859761105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859761105\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3 - 4.24i)T \) |
| good | 5 | \( 1 - 5.65iT - 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 - 30T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 25.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 144T + 1.21e4T^{2} \) |
| 29 | \( 1 + 5.65iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 72T + 5.06e4T^{2} \) |
| 41 | \( 1 - 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 229. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 576T + 1.03e5T^{2} \) |
| 53 | \( 1 + 514. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 414T + 2.05e5T^{2} \) |
| 61 | \( 1 + 504T + 2.26e5T^{2} \) |
| 67 | \( 1 - 789. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 720T + 3.57e5T^{2} \) |
| 73 | \( 1 - 178T + 3.89e5T^{2} \) |
| 79 | \( 1 + 967. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 438T + 5.71e5T^{2} \) |
| 89 | \( 1 + 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 650T + 9.12e5T^{2} \) |
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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
−10.13899214341900857955868272112, −9.269344872992837976155001630935, −8.666636444021353293013619027303, −7.950495980934709103522300265100, −6.55109439563130940834450658891, −5.87483246301908850082338604726, −4.68146771312191446444942494439, −3.63609502191510950065329334440, −2.88414852632983942198080465081, −1.53821500099580977705815329629,
0.811579040061299962554660406386, 1.41622080603324159481921928945, 3.06772048897158497848579256964, 3.91034923086502229398601516015, 5.09953951291311930579745150150, 6.46675890895733500390434668570, 6.95045361326107877536244052875, 7.922814700948561537578784069393, 8.926741821150822672903021602055, 9.168106010111511902872344041268
