L(s) = 1 | + (−1.81 + 1.04i)5-s + (−0.143 − 0.0829i)7-s + (−0.784 + 1.35i)11-s + (−1.93 − 3.34i)13-s + 5.27i·17-s − 8.05i·19-s + (−2.67 − 4.63i)23-s + (−0.298 + 0.516i)25-s + (−6.75 − 3.89i)29-s + (2.10 − 1.21i)31-s + 0.348·35-s − 8.53·37-s + (−2.47 + 1.43i)41-s + (3.42 + 1.97i)43-s + (3.68 − 6.38i)47-s + ⋯ |
L(s) = 1 | + (−0.812 + 0.469i)5-s + (−0.0543 − 0.0313i)7-s + (−0.236 + 0.409i)11-s + (−0.535 − 0.928i)13-s + 1.27i·17-s − 1.84i·19-s + (−0.557 − 0.966i)23-s + (−0.0596 + 0.103i)25-s + (−1.25 − 0.724i)29-s + (0.378 − 0.218i)31-s + 0.0588·35-s − 1.40·37-s + (−0.387 + 0.223i)41-s + (0.521 + 0.301i)43-s + (0.537 − 0.931i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.105313 - 0.299234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105313 - 0.299234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.81 - 1.04i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.143 + 0.0829i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.784 - 1.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.93 + 3.34i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.27iT - 17T^{2} \) |
| 19 | \( 1 + 8.05iT - 19T^{2} \) |
| 23 | \( 1 + (2.67 + 4.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.75 + 3.89i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.10 + 1.21i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + (2.47 - 1.43i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.68 + 6.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.40iT - 53T^{2} \) |
| 59 | \( 1 + (5.49 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.11 - 12.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.45 - 0.841i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + (-6.31 - 3.64i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.35 - 2.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.40iT - 89T^{2} \) |
| 97 | \( 1 + (-0.903 + 1.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.00972917829112270944335879022, −8.908811877627962813788740897536, −8.025774883285868536600153889896, −7.34986412807949759451908757436, −6.51087498946561648844261005123, −5.36471997656578037283028128174, −4.34875841388226733125200492276, −3.36573453048972755788362049924, −2.23905862412219906545135829185, −0.14756751368874980584200165406,
1.69490276380711529514913975513, 3.25956506762951283826185106601, 4.17020467927394415253348141636, 5.14959670845423032072405401780, 6.09894241613494444020858137405, 7.36606040775599272071971489336, 7.80239111756923778786327859224, 8.870820845770140513591561393367, 9.531582128948814027140478586649, 10.49842010445527831352367182617