| L(s) = 1 | + (−2.54 − 1.59i)3-s + (−4.96 − 0.625i)5-s + 5.93i·7-s + (3.92 + 8.10i)9-s + 5.99·11-s + 15.6·13-s + (11.6 + 9.49i)15-s − 14.2·17-s − 29.6i·19-s + (9.46 − 15.0i)21-s − 21.5·23-s + (24.2 + 6.20i)25-s + (2.93 − 26.8i)27-s + 18.1·29-s + 8.55·31-s + ⋯ |
| L(s) = 1 | + (−0.847 − 0.531i)3-s + (−0.992 − 0.125i)5-s + 0.848i·7-s + (0.435 + 0.900i)9-s + 0.545·11-s + 1.20·13-s + (0.774 + 0.632i)15-s − 0.836·17-s − 1.55i·19-s + (0.450 − 0.718i)21-s − 0.935·23-s + (0.968 + 0.248i)25-s + (0.108 − 0.994i)27-s + 0.625·29-s + 0.276·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1034964870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1034964870\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.54 + 1.59i)T \) |
| 5 | \( 1 + (4.96 + 0.625i)T \) |
| good | 7 | \( 1 - 5.93iT - 49T^{2} \) |
| 11 | \( 1 - 5.99T + 121T^{2} \) |
| 13 | \( 1 - 15.6T + 169T^{2} \) |
| 17 | \( 1 + 14.2T + 289T^{2} \) |
| 19 | \( 1 + 29.6iT - 361T^{2} \) |
| 23 | \( 1 + 21.5T + 529T^{2} \) |
| 29 | \( 1 - 18.1T + 841T^{2} \) |
| 31 | \( 1 - 8.55T + 961T^{2} \) |
| 37 | \( 1 + 13.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 93.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 103.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 14.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.9iT - 9.40e3T^{2} \) |
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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
−10.50049545295113460230753915105, −9.089382074071811980773834938999, −8.554726209808674419511745611545, −7.62321256062992964228643858655, −6.64827619227524924385106895734, −6.09480825275642533122755128992, −4.92164207336071753502095831303, −4.15680525797639246538830392355, −2.75915993192825246949016272945, −1.31320257443818808325904717777,
0.04289311993839669206207585804, 1.35892009699461026741331173304, 3.62963871250089839368949909385, 3.91256177622737885625892634200, 4.90165264129610941637654077546, 6.21494402879667842006485213926, 6.68414286004147371830373852353, 7.82490334611888783182990238875, 8.561715595716921817575161220923, 9.655684870060341138848865877471
