L(s) = 1 | − 14.6i·2-s − 54.7i·3-s − 86.1·4-s − 801.·6-s + 343i·7-s − 612. i·8-s − 815.·9-s − 6.47e3·11-s + 4.71e3i·12-s − 1.16e4i·13-s + 5.01e3·14-s − 1.99e4·16-s − 1.34e4i·17-s + 1.19e4i·18-s − 3.49e4·19-s + ⋯ |
L(s) = 1 | − 1.29i·2-s − 1.17i·3-s − 0.672·4-s − 1.51·6-s + 0.377i·7-s − 0.423i·8-s − 0.373·9-s − 1.46·11-s + 0.788i·12-s − 1.47i·13-s + 0.488·14-s − 1.22·16-s − 0.664i·17-s + 0.482i·18-s − 1.16·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.4691772042\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4691772042\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - 343iT \) |
good | 2 | \( 1 + 14.6iT - 128T^{2} \) |
| 3 | \( 1 + 54.7iT - 2.18e3T^{2} \) |
| 11 | \( 1 + 6.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.16e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.34e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.78e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 2.21e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.32e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.22e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.10e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.40e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.06e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 4.51e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.31e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.26e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.99e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.72e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.38e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 7.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.38e6iT - 8.07e13T^{2} \) |
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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
−10.59910036649227003460681984197, −9.883667352151273925243704374136, −8.345260973671947250252093844354, −7.52110851168903530837009294803, −6.26625479956127827579251344849, −4.91231442150107180521269068031, −3.07397692342558551037510063975, −2.38138020682766066257039338620, −1.16235573075835217919670434175, −0.12300511325538373080072329152,
2.34535579846977756801015972934, 4.20630078266436256923475806695, 4.82485912312417685325514718931, 6.10030667447000794786714013841, 7.08475537155636107006746575027, 8.265180891275219169559217624113, 9.064830808137545178437373181929, 10.42208863852688946851947795079, 10.85549075894255542290481289273, 12.43680745077416922800832362932