| L(s) = 1 | + (0.937 + 1.76i)2-s + (0.677 − 2.52i)3-s + (−2.24 + 3.31i)4-s + (−0.838 + 4.92i)5-s + (5.09 − 1.17i)6-s + (5.54 + 4.27i)7-s + (−7.95 − 0.853i)8-s + (1.86 + 1.07i)9-s + (−9.49 + 3.14i)10-s + (−0.558 + 0.322i)11-s + (6.85 + 7.90i)12-s + (1.26 − 1.26i)13-s + (−2.36 + 13.7i)14-s + (11.8 + 5.45i)15-s + (−5.95 − 14.8i)16-s + (−5.07 + 18.9i)17-s + ⋯ |
| L(s) = 1 | + (0.468 + 0.883i)2-s + (0.225 − 0.842i)3-s + (−0.560 + 0.828i)4-s + (−0.167 + 0.985i)5-s + (0.849 − 0.195i)6-s + (0.791 + 0.611i)7-s + (−0.994 − 0.106i)8-s + (0.207 + 0.119i)9-s + (−0.949 + 0.314i)10-s + (−0.0507 + 0.0292i)11-s + (0.571 + 0.658i)12-s + (0.0970 − 0.0970i)13-s + (−0.168 + 0.985i)14-s + (0.792 + 0.363i)15-s + (−0.371 − 0.928i)16-s + (−0.298 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00674 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00674 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31752 + 1.30867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31752 + 1.30867i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.937 - 1.76i)T \) |
| 5 | \( 1 + (0.838 - 4.92i)T \) |
| 7 | \( 1 + (-5.54 - 4.27i)T \) |
| good | 3 | \( 1 + (-0.677 + 2.52i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (0.558 - 0.322i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.26 + 1.26i)T - 169iT^{2} \) |
| 17 | \( 1 + (5.07 - 18.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-22.5 - 13.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.50 + 31.7i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 39.4iT - 841T^{2} \) |
| 31 | \( 1 + (14.1 + 24.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-4.16 - 15.5i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 37.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-34.6 + 34.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (17.3 + 64.7i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-14.4 + 53.8i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (48.7 - 28.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-52.9 - 30.5i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (0.166 - 0.620i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 61.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-57.8 - 15.5i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (43.1 - 74.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-4.14 - 4.14i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (26.4 - 45.8i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (45.6 + 45.6i)T + 9.40e3iT^{2} \) |
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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
−13.37274719265176339753391917058, −12.35810676601186805883645313750, −11.48820883226531836180003890193, −10.04428847241376589573720454329, −8.340592633717101491618692192247, −7.80472015093328683351584983743, −6.72319369860463703145882016415, −5.73239848333352271967069980938, −4.09677625067188484720512714587, −2.36863693902608951044438564597,
1.23556983804379426611184980128, 3.45540818747406291705692938332, 4.60585994201992085844464130020, 5.25689844158949553903374063885, 7.43641018908604136604089473338, 9.041255995737824966722803542842, 9.514465404622227873378524320951, 10.77698220382782080502949973647, 11.57539260180635389835545082215, 12.60210067412679719074788167049
