| L(s) = 1 | + 2.23i·5-s − 7.68i·7-s − 8.06i·11-s + 14.0·13-s − 21.5i·17-s − 5.94i·19-s + (16.1 + 16.3i)23-s − 5.00·25-s − 9.04·29-s − 34.1·31-s + 17.1·35-s − 55.1i·37-s − 42.2·41-s − 37.9i·43-s + 70.5·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 1.09i·7-s − 0.733i·11-s + 1.07·13-s − 1.26i·17-s − 0.312i·19-s + (0.703 + 0.710i)23-s − 0.200·25-s − 0.312·29-s − 1.10·31-s + 0.490·35-s − 1.49i·37-s − 1.03·41-s − 0.881i·43-s + 1.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.536922589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536922589\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-16.1 - 16.3i)T \) |
| good | 7 | \( 1 + 7.68iT - 49T^{2} \) |
| 11 | \( 1 + 8.06iT - 121T^{2} \) |
| 13 | \( 1 - 14.0T + 169T^{2} \) |
| 17 | \( 1 + 21.5iT - 289T^{2} \) |
| 19 | \( 1 + 5.94iT - 361T^{2} \) |
| 29 | \( 1 + 9.04T + 841T^{2} \) |
| 31 | \( 1 + 34.1T + 961T^{2} \) |
| 37 | \( 1 + 55.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 37.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 70.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 5.95T + 3.48e3T^{2} \) |
| 61 | \( 1 - 25.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 21.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 85.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 75.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 53.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 56.0iT - 9.40e3T^{2} \) |
| 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
−7.73659004170068809667692874030, −7.27915363698477908564227171955, −6.66194353385187267239559568901, −5.72797049446385777777306734098, −5.07665008021397479047706470719, −3.84229507697662630198754941664, −3.56033758523348027570359734805, −2.47030849769383571797714954663, −1.20471829809693346361795726656, −0.33515548291593813232818371529,
1.28446272386937234158358374992, 2.01670593847620959344181931350, 3.09285584606828133066023153989, 4.00439786149868039442718477964, 4.80499004491417599127163619115, 5.66644920876844419173652987068, 6.16770446203002674870152896726, 7.01098848276993115967611416720, 7.993331698656449087964390087365, 8.622062222822698776662791556341
