| L(s) = 1 | + 199. i·5-s − 19.6i·7-s − 924.·11-s − 1.55e3i·13-s − 5.14e3·17-s + 1.69e3·19-s − 1.92e4i·23-s − 2.40e4·25-s − 1.65e4i·29-s + 7.55e3i·31-s + 3.91e3·35-s + 2.89e4i·37-s + 5.21e4·41-s − 5.89e3·43-s − 6.44e4i·47-s + ⋯ |
| L(s) = 1 | + 1.59i·5-s − 0.0573i·7-s − 0.694·11-s − 0.705i·13-s − 1.04·17-s + 0.247·19-s − 1.57i·23-s − 1.53·25-s − 0.680i·29-s + 0.253i·31-s + 0.0913·35-s + 0.571i·37-s + 0.756·41-s − 0.0741·43-s − 0.620i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.140111022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140111022\) |
| \(L(4)\) |
|
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 | \( 1 - 199. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 19.6iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 924.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.55e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.14e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.69e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.92e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.65e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 7.55e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.89e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 5.21e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 5.89e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + 6.44e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.97e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.42e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 9.64e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 7.52e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.56e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.85e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.42e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 9.29e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.34e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.43e5T + 8.32e11T^{2} \) |
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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
−10.61618275307730788322636406796, −10.02931188880507986685955952839, −8.627202518894371810793176578316, −7.57741977465943521080968304160, −6.74431053293805587827827249863, −5.83829063821957298764500413176, −4.39901808625621284685089517762, −3.03881369193657409999809007841, −2.32454858319437693379362575975, −0.31546089857718901907334876481,
0.986859395062464506460861654539, 2.15172515902447608550998120405, 3.89332262423651739082680355780, 4.90669261669277072991367815252, 5.68484916439075230320847761220, 7.12829998153716156255133503067, 8.207377397901799029750394602767, 9.031737507820534118785708181186, 9.696834358311238477693830940759, 11.06823009328837420668064635415
