| L(s) = 1 | − 2·2-s + 4·4-s + 5i·5-s + 4.15·7-s − 8·8-s − 10i·10-s + 13.6i·11-s − 65.3i·13-s − 8.30·14-s + 16·16-s + 44.6i·17-s + (29.3 + 77.4i)19-s + 20i·20-s − 27.2i·22-s − 76.5i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447i·5-s + 0.224·7-s − 0.353·8-s − 0.316i·10-s + 0.373i·11-s − 1.39i·13-s − 0.158·14-s + 0.250·16-s + 0.637i·17-s + (0.354 + 0.935i)19-s + 0.223i·20-s − 0.264i·22-s − 0.694i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.559i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5867593490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5867593490\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| 19 | \( 1 + (-29.3 - 77.4i)T \) |
| good | 7 | \( 1 - 4.15T + 343T^{2} \) |
| 11 | \( 1 - 13.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 65.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 44.6iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 76.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 85.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 367. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 29.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 217.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 613. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 514. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 233.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 42.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 41.1iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 600.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 554. iT - 9.12e5T^{2} \) |
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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
−9.336230801547887071662058195799, −8.412963878071724037344569927470, −7.73465815646760253375408665734, −7.19393708826785922096802080655, −6.04098410575763234421312875418, −5.52784917515102136785623573632, −4.18501558846577891398705837589, −3.20407970311095519804470629736, −2.22619982427653112217380837654, −1.11367173413494443670821622717,
0.17330850013135298693173724892, 1.32432622643222261251096189259, 2.27386967654589751157228864076, 3.47530838490169579500136086758, 4.55967573093771665944445354603, 5.39805907213875784878524901946, 6.41917063733571046127564223105, 7.19467536397261150479666798418, 7.88353010691662908417303763237, 8.939334290456865969027309526594
