| L(s) = 1 | + (1.29 − 0.561i)2-s + (1.55 + 2.47i)3-s + (1.36 − 1.45i)4-s + (−2.53 − 2.02i)5-s + (3.40 + 2.33i)6-s + (−2.96 − 0.675i)7-s + (0.957 − 2.66i)8-s + (−2.39 + 4.96i)9-s + (−4.42 − 1.19i)10-s + (−0.356 + 1.01i)11-s + (5.72 + 1.11i)12-s + (1.93 + 4.02i)13-s + (−4.22 + 0.785i)14-s + (1.05 − 9.40i)15-s + (−0.252 − 3.99i)16-s + (−1.15 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.917 − 0.397i)2-s + (0.896 + 1.42i)3-s + (0.684 − 0.729i)4-s + (−1.13 − 0.904i)5-s + (1.38 + 0.952i)6-s + (−1.11 − 0.255i)7-s + (0.338 − 0.940i)8-s + (−0.797 + 1.65i)9-s + (−1.39 − 0.379i)10-s + (−0.107 + 0.307i)11-s + (1.65 + 0.322i)12-s + (0.537 + 1.11i)13-s + (−1.12 + 0.210i)14-s + (0.273 − 2.42i)15-s + (−0.0631 − 0.998i)16-s + (−0.279 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72555 + 0.0862506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72555 + 0.0862506i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.561i)T \) |
| 29 | \( 1 + (-4.63 + 2.74i)T \) |
| good | 3 | \( 1 + (-1.55 - 2.47i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (2.53 + 2.02i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (2.96 + 0.675i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (0.356 - 1.01i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-1.93 - 4.02i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.15 + 1.15i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.711 + 0.447i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-4.45 + 3.55i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.542 - 4.81i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (4.22 - 1.47i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (3.70 - 3.70i)T - 41iT^{2} \) |
| 43 | \( 1 + (-8.74 - 0.985i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (6.77 + 2.37i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-1.23 + 1.55i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 3.13iT - 59T^{2} \) |
| 61 | \( 1 + (1.32 - 0.832i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (-1.37 - 0.661i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (3.81 - 1.83i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.65 + 0.298i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (15.4 - 5.40i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 2.97i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (7.67 - 0.865i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-5.71 - 3.59i)T + (42.0 + 87.3i)T^{2} \) |
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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
−13.61397501560921297607878822299, −12.64393692227857535553375025489, −11.54827240799599742581499474066, −10.43731202140007507804100693249, −9.429727053565068981803358217100, −8.527965419325627156245637757864, −6.79146392087623724455497124812, −4.78806810282015496220718696694, −4.13931459023658538081602908391, −3.10223900569340674614162954914,
2.87225390958265654901687745234, 3.50245019873446813135702689540, 6.04513194684536282836369434611, 6.97323903014755883087585178672, 7.71979047849887517943787519389, 8.668352182798795608482741820265, 10.79652128795704218079467855779, 11.95108730591512427501863279727, 12.82483000254974877263859638861, 13.38929446850250224964490912476
