| L(s) = 1 | − 2.71i·2-s − 3.37·4-s − 6.95i·5-s + 4.31·7-s − 1.69i·8-s − 18.8·10-s + 10.6i·11-s + 12.5·13-s − 11.7i·14-s − 18.1·16-s − 32.6i·17-s − 15.7·19-s + 23.4i·20-s + 28.9·22-s + 11.1i·23-s + ⋯ |
| L(s) = 1 | − 1.35i·2-s − 0.844·4-s − 1.39i·5-s + 0.616·7-s − 0.211i·8-s − 1.88·10-s + 0.970i·11-s + 0.962·13-s − 0.837i·14-s − 1.13·16-s − 1.91i·17-s − 0.827·19-s + 1.17i·20-s + 1.31·22-s + 0.486i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.635345128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635345128\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + 11.2T \) |
| good | 2 | \( 1 + 2.71iT - 4T^{2} \) |
| 5 | \( 1 + 6.95iT - 25T^{2} \) |
| 7 | \( 1 - 4.31T + 49T^{2} \) |
| 11 | \( 1 - 10.6iT - 121T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 17 | \( 1 + 32.6iT - 289T^{2} \) |
| 19 | \( 1 + 15.7T + 361T^{2} \) |
| 23 | \( 1 - 11.1iT - 529T^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 + 12.3T + 961T^{2} \) |
| 37 | \( 1 + 7.23T + 1.36e3T^{2} \) |
| 41 | \( 1 - 3.87iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.07iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.76iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 64.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 52.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 112.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 56.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 76.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.6T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.396724763174436065396890856555, −8.542394927443682465879351780118, −7.67564374586207270480849670544, −6.54429792357984168860715382622, −5.15038867886138407558530447306, −4.60848923367291495124610756417, −3.73752102087974554679355750983, −2.35346278100669030057242615745, −1.48967271410150622767558596411, −0.48790604747349943841761029860,
1.79359316222316187914299974107, 3.17893655032877026743831072669, 4.14759924669502464800151641932, 5.46644499267444938279898639969, 6.24368384759746593133626777127, 6.62105260660679559776127491443, 7.58921960678589559955654187732, 8.455116276116148797437164991012, 8.723622179469313920907858059109, 10.36235670746472898078988165801
