| L(s) = 1 | + (−0.976 + 0.214i)2-s + (−0.840 − 1.51i)3-s + (0.907 − 0.419i)4-s + (4.02 + 1.11i)5-s + (1.14 + 1.29i)6-s + (1.69 − 1.60i)7-s + (−0.796 + 0.605i)8-s + (−1.58 + 2.54i)9-s + (−4.17 − 0.226i)10-s + (1.17 + 1.38i)11-s + (−1.39 − 1.02i)12-s + (1.59 + 4.74i)13-s + (−1.30 + 1.92i)14-s + (−1.69 − 7.03i)15-s + (0.647 − 0.762i)16-s + (−0.854 + 0.902i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.152i)2-s + (−0.485 − 0.874i)3-s + (0.453 − 0.209i)4-s + (1.80 + 0.499i)5-s + (0.468 + 0.529i)6-s + (0.639 − 0.606i)7-s + (−0.281 + 0.213i)8-s + (−0.528 + 0.848i)9-s + (−1.31 − 0.0715i)10-s + (0.353 + 0.416i)11-s + (−0.403 − 0.294i)12-s + (0.443 + 1.31i)13-s + (−0.349 + 0.515i)14-s + (−0.436 − 1.81i)15-s + (0.161 − 0.190i)16-s + (−0.207 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18794 - 0.166032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18794 - 0.166032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.976 - 0.214i)T \) |
| 3 | \( 1 + (0.840 + 1.51i)T \) |
| 59 | \( 1 + (1.53 + 7.52i)T \) |
| good | 5 | \( 1 + (-4.02 - 1.11i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 1.60i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 1.38i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 4.74i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (0.854 - 0.902i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.45 + 3.64i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (4.79 - 0.521i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-1.57 + 7.15i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-5.78 - 2.30i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.421 + 0.554i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-0.921 + 8.47i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (4.70 + 3.99i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-1.02 - 3.69i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (13.1 - 0.714i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.549 - 2.49i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (4.40 + 5.79i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (4.30 - 1.19i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (11.9 + 8.08i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 4.67i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-0.807 - 1.52i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-2.93 - 0.646i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (-0.146 + 0.0994i)T + (35.9 - 90.1i)T^{2} \) |
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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
−11.21265560690959447744516343670, −10.52141006860402312536818723504, −9.628750747751209943577915321927, −8.694110940074911955937764240033, −7.46484741134084039245114801096, −6.50795570063292366787011783604, −6.14542260624288218603800107229, −4.71520256885525419973746634779, −2.26629748389117465279011290149, −1.52508911369001556567166700219,
1.42182943455909930490248188923, 2.95404365875219663144175780432, 4.79357393145938394581732562879, 5.80066680798234790406651339542, 6.25087874809475890505743991563, 8.296678217245513474075313544352, 8.830604709587368824872684236043, 9.893121653453464796134497023458, 10.25748709575694283875872526287, 11.24658771770615843115324983081
