| L(s) = 1 | + (−0.540 − 0.841i)2-s + (0.0700 + 1.73i)3-s + (−0.415 + 0.909i)4-s + (−3.21 + 0.943i)5-s + (1.41 − 0.994i)6-s + (−1.80 + 1.56i)7-s + (0.989 − 0.142i)8-s + (−2.99 + 0.242i)9-s + (2.53 + 2.19i)10-s + (0.146 + 0.0943i)11-s + (−1.60 − 0.655i)12-s + (0.435 − 0.502i)13-s + (2.29 + 0.673i)14-s + (−1.85 − 5.49i)15-s + (−0.654 − 0.755i)16-s + (1.46 + 3.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (0.0404 + 0.999i)3-s + (−0.207 + 0.454i)4-s + (−1.43 + 0.421i)5-s + (0.578 − 0.406i)6-s + (−0.683 + 0.591i)7-s + (0.349 − 0.0503i)8-s + (−0.996 + 0.0808i)9-s + (0.800 + 0.693i)10-s + (0.0442 + 0.0284i)11-s + (−0.462 − 0.189i)12-s + (0.120 − 0.139i)13-s + (0.613 + 0.180i)14-s + (−0.479 − 1.41i)15-s + (−0.163 − 0.188i)16-s + (0.354 + 0.775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287687 + 0.427070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287687 + 0.427070i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 3 | \( 1 + (-0.0700 - 1.73i)T \) |
| 23 | \( 1 + (-2.56 - 4.05i)T \) |
| good | 5 | \( 1 + (3.21 - 0.943i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.80 - 1.56i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.146 - 0.0943i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.435 + 0.502i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.46 - 3.19i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-6.08 - 2.78i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (5.54 - 2.53i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.22 + 8.55i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.59 - 8.84i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.791 + 2.69i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (5.73 + 0.824i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 7.88iT - 47T^{2} \) |
| 53 | \( 1 + (-7.73 - 8.93i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.68 - 6.65i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.619 - 0.0891i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.04 - 6.29i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (2.12 + 3.31i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.52 - 5.53i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.93 + 3.40i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-11.4 - 3.34i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.16 + 15.0i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.10 + 3.76i)T + (-81.6 + 52.4i)T^{2} \) |
| 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
−13.35203103242392038750577068280, −11.96186287828523588380400763764, −11.54739184938148497717780157648, −10.42733427454645250899990829290, −9.520884035685781149649475977734, −8.459291322958385232547885293685, −7.41580006776251228937541864170, −5.56931850114443235561850754952, −3.86322877543449006011483223901, −3.18752177695757134439454584521,
0.60102830555526306943744187145, 3.44101748572732251403013816507, 5.15928054510297351222088236878, 6.85438032192300310615161827224, 7.38315655554432107038726204652, 8.377376606795605614859454097000, 9.420400681608262910343338340233, 11.04164470441410858392503567956, 11.93169230233870548775171023429, 12.89053734148107053842941238620
