L(s) = 1 | + 4·2-s + (−15.5 + 1.23i)3-s + 16·4-s − 14.2i·5-s + (−62.1 + 4.95i)6-s + 129.·7-s + 64·8-s + (239. − 38.5i)9-s − 56.8i·10-s − 605.·11-s + (−248. + 19.8i)12-s + 1.14e3i·13-s + 516.·14-s + (17.6 + 220. i)15-s + 256·16-s − 1.59e3i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.996 + 0.0795i)3-s + 0.5·4-s − 0.254i·5-s + (−0.704 + 0.0562i)6-s + 0.995·7-s + 0.353·8-s + (0.987 − 0.158i)9-s − 0.179i·10-s − 1.50·11-s + (−0.498 + 0.0397i)12-s + 1.87i·13-s + 0.703·14-s + (0.0202 + 0.253i)15-s + 0.250·16-s − 1.33i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.044906909\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.044906909\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + (15.5 - 1.23i)T \) |
| 59 | \( 1 + (9.61e3 - 2.49e4i)T \) |
good | 5 | \( 1 + 14.2iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 129.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 605.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.14e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.59e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.10e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.12e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 8.51e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 783. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.29e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 4.46e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.51e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.03e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 3.78e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.74e5iT - 8.58e9T^{2} \) |
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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
−11.12122174881950616757820497874, −10.21685515261935396596883658838, −9.066548843003284930978563341615, −7.64484725497655586890418442274, −6.96616840720682296226642371885, −5.64302571928564424806088013485, −4.94645704840512431795921451806, −4.25450293479117463120039483287, −2.44550347143793811964617546105, −1.15782591046073374970374056607,
0.50091506522894148921083178916, 1.96959335440273203568352049094, 3.37893744480291172508612639646, 4.84403246913323140039313003628, 5.41027979675558282021858376102, 6.25496981477897345998148017299, 7.75261147936476674318699295077, 7.996169295317597473179781942944, 10.15335968738707454703815015637, 10.61161419018316456149484174947