| L(s) = 1 | + (1.87 − 5.15i)3-s + (0.722 + 4.09i)5-s + (2.33 − 4.05i)7-s + (−16.1 − 13.5i)9-s + (−4.49 − 7.77i)11-s + (7.53 + 20.7i)13-s + (22.4 + 3.96i)15-s + (2.56 − 2.15i)17-s + (15.9 + 10.3i)19-s + (−16.4 − 19.6i)21-s + (−6.25 + 35.4i)23-s + (7.22 − 2.62i)25-s + (−57.1 + 32.9i)27-s + (−1.48 + 1.76i)29-s + (−4.89 − 2.82i)31-s + ⋯ |
| L(s) = 1 | + (0.624 − 1.71i)3-s + (0.144 + 0.819i)5-s + (0.334 − 0.578i)7-s + (−1.79 − 1.50i)9-s + (−0.408 − 0.707i)11-s + (0.579 + 1.59i)13-s + (1.49 + 0.264i)15-s + (0.150 − 0.126i)17-s + (0.837 + 0.547i)19-s + (−0.784 − 0.935i)21-s + (−0.271 + 1.54i)23-s + (0.288 − 0.105i)25-s + (−2.11 + 1.22i)27-s + (−0.0511 + 0.0610i)29-s + (−0.157 − 0.0910i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.20763 - 0.897058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20763 - 0.897058i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-15.9 - 10.3i)T \) |
| good | 3 | \( 1 + (-1.87 + 5.15i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (-0.722 - 4.09i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 4.05i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.49 + 7.77i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.53 - 20.7i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-2.56 + 2.15i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (6.25 - 35.4i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (1.48 - 1.76i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (4.89 + 2.82i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 62.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (23.6 - 64.9i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (4.30 + 24.4i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (3.19 + 2.67i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (58.6 + 10.3i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (60.9 + 72.6i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (0.775 - 4.39i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (22.5 - 26.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (8.23 - 1.45i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (58.9 + 21.4i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-24.3 + 66.9i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-24.7 + 42.8i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (0.257 + 0.706i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-24.0 - 28.6i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−13.97331061704410229701127012566, −13.29729506884860335842090574832, −11.88901664730603731638458604537, −11.05918680041985739388465022184, −9.261045378481499307130842311618, −7.922818719046233333282569533132, −7.12484545257092565346372034318, −6.06957146490463803088457265360, −3.32593198056242598183533312988, −1.62950150231237325224137629851,
2.97790064689690504419152876025, 4.65744402184753237280478850126, 5.44885913329425229386741176719, 8.134023561328142590857181720066, 8.866122255974965980395317437008, 9.975497299913628273780152736168, 10.79194028094658230103665391162, 12.33206797954011243799956062271, 13.55835710210550410849563692687, 14.87902277654915544679127882164
