| 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 | \( 1 + (1.10 - 3.76i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−12.89053734148107053842941238620, −11.93169230233870548775171023429, −11.04164470441410858392503567956, −9.420400681608262910343338340233, −8.377376606795605614859454097000, −7.38315655554432107038726204652, −6.85438032192300310615161827224, −5.15928054510297351222088236878, −3.44101748572732251403013816507, −0.60102830555526306943744187145,
3.18752177695757134439454584521, 3.86322877543449006011483223901, 5.56931850114443235561850754952, 7.41580006776251228937541864170, 8.459291322958385232547885293685, 9.520884035685781149649475977734, 10.42733427454645250899990829290, 11.54739184938148497717780157648, 11.96186287828523588380400763764, 13.35203103242392038750577068280
