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
−11.86586236610939113504459530677, −10.56655938448393779726388594909, −10.12429556056137588657273344798, −8.857349822886173038365560604987, −7.79800925107036395633258647588, −7.24408358688003509157871583403, −5.02387700397642519750228932324, −3.99952181393726488108461680756, −2.52935640703144750636400486476, −0.22735496729930546521910322768,
2.74904789831606887210176424309, 4.62069493372382132835561745758, 5.79971954218215620874620689117, 7.01948256460463291397360487302, 7.77335893817623211914734177740, 8.779880487750386560501754435362, 9.777952017246394902184047695306, 10.85931619725685103496820546789, 11.65625751659639797458822361157, 12.97665741228564896699887343772