L(s) = 1 | + (−1.40 − 0.111i)2-s + (−1.71 + 0.231i)3-s + (1.97 + 0.313i)4-s + (1.74 − 3.01i)5-s + (2.44 − 0.136i)6-s + (1.80 − 1.04i)7-s + (−2.75 − 0.660i)8-s + (2.89 − 0.795i)9-s + (−2.79 + 4.06i)10-s + (−0.116 + 0.0675i)11-s + (−3.46 − 0.0795i)12-s + (−2.63 − 1.52i)13-s + (−2.66 + 1.27i)14-s + (−2.29 + 5.58i)15-s + (3.80 + 1.23i)16-s + 4.19i·17-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0785i)2-s + (−0.990 + 0.133i)3-s + (0.987 + 0.156i)4-s + (0.779 − 1.35i)5-s + (0.998 − 0.0556i)6-s + (0.683 − 0.394i)7-s + (−0.972 − 0.233i)8-s + (0.964 − 0.265i)9-s + (−0.883 + 1.28i)10-s + (−0.0352 + 0.0203i)11-s + (−0.999 − 0.0229i)12-s + (−0.731 − 0.422i)13-s + (−0.712 + 0.339i)14-s + (−0.591 + 1.44i)15-s + (0.950 + 0.309i)16-s + 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.502630 - 0.224450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.502630 - 0.224450i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.111i)T \) |
| 3 | \( 1 + (1.71 - 0.231i)T \) |
good | 5 | \( 1 + (-1.74 + 3.01i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 1.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.116 - 0.0675i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.63 + 1.52i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.19iT - 17T^{2} \) |
| 19 | \( 1 - 0.919T + 19T^{2} \) |
| 23 | \( 1 + (-0.689 + 1.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.24 - 7.34i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.39 + 2.53i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.61iT - 37T^{2} \) |
| 41 | \( 1 + (-1.79 - 1.03i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.41 - 9.37i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.205 + 0.356i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.968T + 53T^{2} \) |
| 59 | \( 1 + (-3.88 - 2.24i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.44 - 4.29i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.15 + 5.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.06T + 73T^{2} \) |
| 79 | \( 1 + (-10.8 + 6.27i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.23 + 3.02i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 8.35iT - 89T^{2} \) |
| 97 | \( 1 + (0.477 + 0.826i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.75886721933899907486941920718, −12.93588000341063768635184315643, −12.24385270547313969971813024573, −10.94373155242871265975861597831, −10.03390696439833679762292967525, −8.944653791497725485395748542111, −7.63354824851523704054865937090, −6.03107002593195687650788738258, −4.80829609516533304167432200730, −1.35701781222327843385092819165,
2.28951442477191674827549885237, 5.44839105589224587843496515525, 6.65495640800549514971201894843, 7.52370383338204410020095903893, 9.426376328608089773669044063070, 10.36984997378611447026363969090, 11.26572364968157138743612111986, 12.06751468291407635884503497985, 13.92835048467479007784212461946, 14.97088758574809157250808700697