L(s) = 1 | + (−0.820 + 1.15i)2-s + (−0.654 − 1.88i)4-s + (−1.97 + 1.14i)5-s + (−0.907 + 1.57i)7-s + (2.71 + 0.795i)8-s + (0.306 − 3.21i)10-s + (−4.24 − 2.44i)11-s + (−4.00 + 2.31i)13-s + (−1.06 − 2.33i)14-s + (−3.14 + 2.47i)16-s − 1.92·17-s − 2.12i·19-s + (3.44 + 2.98i)20-s + (6.30 − 2.87i)22-s + (1.15 + 2.00i)23-s + ⋯ |
L(s) = 1 | + (−0.579 + 0.814i)2-s + (−0.327 − 0.944i)4-s + (−0.883 + 0.510i)5-s + (−0.343 + 0.594i)7-s + (0.959 + 0.281i)8-s + (0.0968 − 1.01i)10-s + (−1.27 − 0.738i)11-s + (−1.11 + 0.641i)13-s + (−0.285 − 0.624i)14-s + (−0.785 + 0.618i)16-s − 0.467·17-s − 0.488i·19-s + (0.771 + 0.667i)20-s + (1.34 − 0.613i)22-s + (0.241 + 0.418i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0432053 - 0.232585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0432053 - 0.232585i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.820 - 1.15i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.97 - 1.14i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.907 - 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.24 + 2.44i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.00 - 2.31i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 + 2.12iT - 19T^{2} \) |
| 23 | \( 1 + (-1.15 - 2.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.16 - 1.82i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.65 + 4.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.98iT - 37T^{2} \) |
| 41 | \( 1 + (-2.36 - 4.09i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.02 + 3.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.95iT - 53T^{2} \) |
| 59 | \( 1 + (-3.05 + 1.76i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.71 + 0.991i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.72 + 4.46i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (4.97 - 8.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.12 - 1.80i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 + (-6.99 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.97665741228564896699887343772, −11.65625751659639797458822361157, −10.85931619725685103496820546789, −9.777952017246394902184047695306, −8.779880487750386560501754435362, −7.77335893817623211914734177740, −7.01948256460463291397360487302, −5.79971954218215620874620689117, −4.62069493372382132835561745758, −2.74904789831606887210176424309,
0.22735496729930546521910322768, 2.52935640703144750636400486476, 3.99952181393726488108461680756, 5.02387700397642519750228932324, 7.24408358688003509157871583403, 7.79800925107036395633258647588, 8.857349822886173038365560604987, 10.12429556056137588657273344798, 10.56655938448393779726388594909, 11.86586236610939113504459530677