| L(s) = 1 | + (1.41 + 0.0961i)2-s + (−1.29 + 3.13i)3-s + (1.98 + 0.271i)4-s + (−0.453 + 0.891i)5-s + (−2.13 + 4.29i)6-s + (0.443 + 1.84i)7-s + (2.76 + 0.573i)8-s + (−6.01 − 6.01i)9-s + (−0.726 + 1.21i)10-s + (0.546 − 0.640i)11-s + (−3.42 + 5.85i)12-s + (2.07 + 3.39i)13-s + (0.447 + 2.64i)14-s + (−2.20 − 2.57i)15-s + (3.85 + 1.07i)16-s + (0.501 + 6.37i)17-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0680i)2-s + (−0.749 + 1.80i)3-s + (0.990 + 0.135i)4-s + (−0.203 + 0.398i)5-s + (−0.870 + 1.75i)6-s + (0.167 + 0.697i)7-s + (0.979 + 0.202i)8-s + (−2.00 − 2.00i)9-s + (−0.229 + 0.383i)10-s + (0.164 − 0.193i)11-s + (−0.988 + 1.69i)12-s + (0.576 + 0.941i)13-s + (0.119 + 0.707i)14-s + (−0.568 − 0.666i)15-s + (0.963 + 0.268i)16-s + (0.121 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.279687 + 2.03341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279687 + 2.03341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0961i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (-1.85 - 6.12i)T \) |
| good | 3 | \( 1 + (1.29 - 3.13i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.443 - 1.84i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-0.546 + 0.640i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 3.39i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.501 - 6.37i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (4.52 + 2.77i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (2.18 + 1.58i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.455 + 5.78i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (0.0388 - 0.119i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.40 + 7.39i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.0547 - 0.345i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.0891 + 0.371i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-3.05 - 0.240i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-3.98 + 5.49i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.403 + 2.54i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (7.68 - 6.56i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-5.36 - 4.57i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (7.74 - 7.74i)T - 73iT^{2} \) |
| 79 | \( 1 + (-13.3 - 5.51i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 + (-10.1 + 2.43i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (2.75 - 2.35i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.79110743770052477991251731161, −10.13487966757379152359701154193, −9.017492417300367831982027145894, −8.268285130030215994679655865889, −6.48773987122504163519952001953, −6.10720289111549327074978225854, −5.20948988275895327353338676060, −4.07697222743532571033242864097, −3.89465674523472376390829622410, −2.43504980659135380679357712260,
0.809634384707404423162976958236, 1.87954809848459468687489219723, 3.19973914086658688300491905223, 4.67823754545820923352680488965, 5.53547191811315489431537626148, 6.29509669833480616498398721619, 7.20412318804332132428413050073, 7.64349154670051100575623744811, 8.582682433188156394484265208405, 10.39944762834342535486343410200
