L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.0795 + 1.73i)3-s + (−0.866 + 0.499i)4-s + (2.02 − 2.02i)5-s + (1.69 − 0.370i)6-s + (3.46 + 0.929i)7-s + (0.707 + 0.707i)8-s + (−2.98 − 0.275i)9-s + (−2.47 − 1.42i)10-s + (−4.05 + 1.08i)11-s + (−0.796 − 1.53i)12-s + (−3.60 + 0.176i)13-s − 3.58i·14-s + (3.33 + 3.65i)15-s + (0.500 − 0.866i)16-s + (−1.72 − 2.99i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.0459 + 0.998i)3-s + (−0.433 + 0.249i)4-s + (0.903 − 0.903i)5-s + (0.690 − 0.151i)6-s + (1.31 + 0.351i)7-s + (0.249 + 0.249i)8-s + (−0.995 − 0.0917i)9-s + (−0.782 − 0.451i)10-s + (−1.22 + 0.327i)11-s + (−0.229 − 0.444i)12-s + (−0.998 + 0.0490i)13-s − 0.959i·14-s + (0.861 + 0.944i)15-s + (0.125 − 0.216i)16-s + (−0.418 − 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.924141 - 0.108079i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.924141 - 0.108079i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.0795 - 1.73i)T \) |
| 13 | \( 1 + (3.60 - 0.176i)T \) |
good | 5 | \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \) |
| 7 | \( 1 + (-3.46 - 0.929i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.05 - 1.08i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.72 + 2.99i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.581 + 2.16i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.51 - 2.62i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.74 - 1.00i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 1.21i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.25 + 4.67i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.05 - 7.67i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.68 - 0.975i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.957 + 0.957i)T + 47iT^{2} \) |
| 53 | \( 1 + 7.22iT - 53T^{2} \) |
| 59 | \( 1 + (2.66 - 9.93i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.137 + 0.237i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.10 + 1.09i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 2.85i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 10.0i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 + (2.58 - 2.58i)T - 83iT^{2} \) |
| 89 | \( 1 + (-9.50 + 2.54i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (2.07 - 7.74i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−14.34999665576343663557214245800, −13.30498566274546126356930094130, −12.05192003280170699633399518703, −11.02973295787310225494602805490, −9.913084502284390938233009411471, −9.113936995245362224996270308275, −7.977261987918138548720620181315, −5.20651396242987531879265507071, −4.83817662665721398111220542249, −2.36436601585895883167443074210,
2.24378862367465598119896816166, 5.17817326656200332643536967820, 6.35727516116318064865580285596, 7.55256144807449842199636548058, 8.328067260244481320194257290623, 10.16372580805401237619137512745, 11.04637294058986745905465341338, 12.57359975520053561972116965673, 13.83985102767818382571752684186, 14.25255225229198875297025957669