L(s) = 1 | − 2.58i·2-s + (−2.60 − 4.49i)3-s + 1.30·4-s − 16.1·5-s + (−11.6 + 6.73i)6-s − 24.0i·8-s + (−13.4 + 23.4i)9-s + 41.7i·10-s + 35.5i·11-s + (−3.39 − 5.87i)12-s − 7.40i·13-s + (41.9 + 72.4i)15-s − 51.8·16-s − 28.9·17-s + (60.5 + 34.8i)18-s + 35.1i·19-s + ⋯ |
L(s) = 1 | − 0.914i·2-s + (−0.500 − 0.865i)3-s + 0.163·4-s − 1.44·5-s + (−0.791 + 0.458i)6-s − 1.06i·8-s + (−0.498 + 0.867i)9-s + 1.31i·10-s + 0.975i·11-s + (−0.0817 − 0.141i)12-s − 0.158i·13-s + (0.722 + 1.24i)15-s − 0.810·16-s − 0.412·17-s + (0.793 + 0.455i)18-s + 0.424i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0449147 + 0.0371328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0449147 + 0.0371328i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.60 + 4.49i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.58iT - 8T^{2} \) |
| 5 | \( 1 + 16.1T + 125T^{2} \) |
| 11 | \( 1 - 35.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 7.40iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 28.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 55.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 68.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 178. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 370.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 272. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 96.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 385. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 125. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 225. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 601.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 327. iT - 9.12e5T^{2} \) |
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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
−12.03986495606974120071939902334, −11.05704510187725099987398764836, −10.23219950365928139264695916528, −8.490553292684599770993499263775, −7.36977684987555882605477428516, −6.66631131754224596105854338093, −4.75943352267055552204410562118, −3.31891214656742935368761598666, −1.71134317599806348531955299144, −0.02921983235725113320737603156,
3.32892649188439304024193537100, 4.59547791092527261817340707172, 5.81907860487782463508866699487, 6.94488502024619539822653203763, 8.078240000377661110817214189692, 8.940253530663185589562710394273, 10.56699939949609591377980249304, 11.49986464782280026948517151253, 11.83825141959965754440248689683, 13.61735150409100246842878023287