L(s) = 1 | + (0.399 − 0.630i)2-s + (1.70 − 1.59i)3-s + (0.614 + 1.30i)4-s + (−0.498 − 2.17i)5-s + (−0.326 − 1.71i)6-s + (−3.50 + 2.54i)7-s + (2.54 + 0.322i)8-s + (0.153 − 2.43i)9-s + (−1.57 − 0.557i)10-s + (−2.53 + 3.98i)11-s + (3.13 + 1.23i)12-s + (0.0635 − 1.00i)13-s + (0.203 + 3.22i)14-s + (−4.33 − 2.91i)15-s + (−0.617 + 0.746i)16-s + (3.03 − 6.44i)17-s + ⋯ |
L(s) = 1 | + (0.282 − 0.445i)2-s + (0.982 − 0.922i)3-s + (0.307 + 0.652i)4-s + (−0.223 − 0.974i)5-s + (−0.133 − 0.698i)6-s + (−1.32 + 0.963i)7-s + (0.901 + 0.113i)8-s + (0.0511 − 0.813i)9-s + (−0.497 − 0.176i)10-s + (−0.763 + 1.20i)11-s + (0.903 + 0.357i)12-s + (0.0176 − 0.280i)13-s + (0.0543 + 0.863i)14-s + (−1.11 − 0.751i)15-s + (−0.154 + 0.186i)16-s + (0.735 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33424 - 0.682868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33424 - 0.682868i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.498 + 2.17i)T \) |
good | 2 | \( 1 + (-0.399 + 0.630i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (-1.70 + 1.59i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (3.50 - 2.54i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (2.53 - 3.98i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (-0.0635 + 1.00i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (-3.03 + 6.44i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (0.225 + 0.212i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.47 - 0.635i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (2.93 + 1.61i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (-0.648 + 1.37i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (3.81 - 4.61i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (1.52 - 0.391i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-2.70 + 8.31i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (3.85 - 0.486i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (-0.548 + 2.87i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (-7.84 - 3.10i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (10.1 + 2.60i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (-3.30 + 1.81i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (-7.73 + 0.977i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-5.38 + 2.13i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (3.01 - 2.82i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (10.6 + 10.0i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (3.89 - 1.54i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-7.39 - 4.06i)T + (51.9 + 81.8i)T^{2} \) |
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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
−12.97821691794801567453908384486, −12.50870547604556347138825406729, −11.81631107312970299885952927183, −9.861016022836821898831184887247, −8.900410551958504543102975358585, −7.81741420976773415117755532106, −7.03594121820158998767398713227, −5.11944372825759485168282539181, −3.25114396783998462020993118400, −2.26019054334143362306572608427,
3.08732231855109843203098099340, 3.93811580906833868071833502028, 5.88267039687788830349061781975, 6.88546392445369521854366480925, 8.112730871525661401474531811285, 9.618378656329811942718623613962, 10.43185566320119322863922977741, 10.89269190107197984571610794060, 12.99750241532304249507848530651, 13.96297607712151389437190211934