| L(s) = 1 | + (−0.777 − 0.499i)2-s + (−0.318 − 2.21i)3-s + (−0.476 − 1.04i)4-s + (−0.959 − 0.281i)5-s + (−0.859 + 1.88i)6-s + (−2.14 + 2.47i)7-s + (−0.413 + 2.87i)8-s + (−1.93 + 0.567i)9-s + (0.605 + 0.698i)10-s + (4.11 − 2.64i)11-s + (−2.15 + 1.38i)12-s + (−3.14 − 3.62i)13-s + (2.90 − 0.852i)14-s + (−0.318 + 2.21i)15-s + (0.257 − 0.297i)16-s + (2.77 − 6.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 − 0.353i)2-s + (−0.183 − 1.27i)3-s + (−0.238 − 0.521i)4-s + (−0.429 − 0.125i)5-s + (−0.350 + 0.768i)6-s + (−0.810 + 0.935i)7-s + (−0.146 + 1.01i)8-s + (−0.643 + 0.189i)9-s + (0.191 + 0.220i)10-s + (1.23 − 0.796i)11-s + (−0.623 + 0.400i)12-s + (−0.872 − 1.00i)13-s + (0.776 − 0.227i)14-s + (−0.0822 + 0.572i)15-s + (0.0644 − 0.0743i)16-s + (0.673 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0985453 - 0.549203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0985453 - 0.549203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-0.819 - 4.72i)T \) |
| good | 2 | \( 1 + (0.777 + 0.499i)T + (0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (0.318 + 2.21i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (2.14 - 2.47i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.11 + 2.64i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.14 + 3.62i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.77 + 6.08i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.76 + 3.86i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.393 + 0.862i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.737 - 5.12i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.22 + 1.24i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-4.53 - 1.33i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.01 - 7.08i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 + (-4.04 + 4.67i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.55 - 5.25i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 9.36i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.94 + 2.53i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.82 - 1.81i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.112 + 0.245i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.918 - 1.06i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.99 + 1.17i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.0112 - 0.0785i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-15.0 - 4.41i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.91351833033102884495150301051, −11.99459686201470615078461351002, −11.32996405301636032202744901455, −9.692387506472150056326414144974, −8.958363052209508311244478858744, −7.67236059365053992335260426675, −6.43923574038896515310446923022, −5.33641810994983715427539165831, −2.78059587014039344302177132214, −0.789108533078811148121337219476,
3.98319407672242377697239991586, 4.11201973230499402086944623199, 6.50502510078186025907654523788, 7.49570838453435459377752195753, 8.935279115872073039491416384092, 9.818868607282584866505292204162, 10.42567476192824425093744512375, 11.97667282430842879578702497879, 12.80990842557170641075656109518, 14.41696425790793772121266198805
