| L(s) = 1 | + (−0.950 − 1.04i)2-s + (−0.854 + 2.58i)3-s + (−0.194 + 1.99i)4-s + (−0.0787 + 0.204i)5-s + (3.52 − 1.56i)6-s + (1.26 − 2.11i)7-s + (2.26 − 1.68i)8-s + (−3.55 − 2.63i)9-s + (0.288 − 0.111i)10-s + (−0.565 − 4.58i)11-s + (−4.98 − 2.20i)12-s + (0.117 − 4.78i)13-s + (−3.42 + 0.681i)14-s + (−0.460 − 0.378i)15-s + (−3.92 − 0.772i)16-s + (4.86 + 5.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.671 − 0.740i)2-s + (−0.493 + 1.49i)3-s + (−0.0970 + 0.995i)4-s + (−0.0351 + 0.0912i)5-s + (1.43 − 0.638i)6-s + (0.479 − 0.799i)7-s + (0.802 − 0.596i)8-s + (−1.18 − 0.879i)9-s + (0.0912 − 0.0352i)10-s + (−0.170 − 1.38i)11-s + (−1.43 − 0.636i)12-s + (0.0325 − 1.32i)13-s + (−0.914 + 0.182i)14-s + (−0.118 − 0.0976i)15-s + (−0.981 − 0.193i)16-s + (1.18 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.833040 - 0.190588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833040 - 0.190588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.950 + 1.04i)T \) |
| good | 3 | \( 1 + (0.854 - 2.58i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (0.0787 - 0.204i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.11i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (0.565 + 4.58i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (-0.117 + 4.78i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (-4.86 - 5.93i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.01 - 0.715i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-3.52 + 7.46i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (-2.07 + 0.574i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (7.54 - 1.50i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.727 + 3.24i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (-8.66 + 7.85i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (3.22 - 6.41i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (-1.90 - 1.01i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (6.50 + 1.80i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 0.0502i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (-4.40 - 3.80i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (6.55 - 0.483i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (-2.25 + 15.2i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (-0.573 - 0.957i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (-16.3 - 4.96i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-3.65 + 16.2i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (-3.06 - 6.48i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (2.13 + 1.42i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.73569164152029454546944501364, −10.41642828313536346326206030809, −9.247765997415491265485358146825, −8.400643294619229646827326172713, −7.60919573336654985120007338890, −5.95459403114385256109201535946, −4.95537362759212029045473694685, −3.77241238910195965604973293067, −3.18637590699882651201353944326, −0.810515357690174009964075640919,
1.26989129870963835080062533247, 2.26632961094228096221138874331, 4.89003193223007834414575257534, 5.54129724057141927242498558644, 6.76111703996046810843455506950, 7.25276601000285446254107976554, 7.929975236187985166538791484236, 9.123199755835563778848749858139, 9.714160160535143788178541965338, 11.22374395937545246341536591843
