| L(s) = 1 | + (0.380 − 2.40i)2-s + (0.532 + 0.271i)3-s + (−3.72 − 1.21i)4-s + (−0.622 + 2.14i)5-s + (0.855 − 1.17i)6-s + (1.17 + 2.30i)7-s + (−2.11 + 4.15i)8-s + (−1.55 − 2.13i)9-s + (4.92 + 2.31i)10-s + (−2.55 − 2.11i)11-s + (−1.65 − 1.65i)12-s + (2.77 + 0.439i)13-s + (5.97 − 1.94i)14-s + (−0.914 + 0.975i)15-s + (2.84 + 2.06i)16-s + (−0.932 + 0.147i)17-s + ⋯ |
| L(s) = 1 | + (0.269 − 1.69i)2-s + (0.307 + 0.156i)3-s + (−1.86 − 0.605i)4-s + (−0.278 + 0.960i)5-s + (0.349 − 0.480i)6-s + (0.443 + 0.869i)7-s + (−0.748 + 1.46i)8-s + (−0.517 − 0.712i)9-s + (1.55 + 0.731i)10-s + (−0.770 − 0.637i)11-s + (−0.478 − 0.478i)12-s + (0.769 + 0.121i)13-s + (1.59 − 0.518i)14-s + (−0.236 + 0.251i)15-s + (0.710 + 0.516i)16-s + (−0.226 + 0.0358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0363 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0363 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.646417 - 0.670334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.646417 - 0.670334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.622 - 2.14i)T \) |
| 11 | \( 1 + (2.55 + 2.11i)T \) |
| good | 2 | \( 1 + (-0.380 + 2.40i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.532 - 0.271i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 2.30i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (-2.77 - 0.439i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.932 - 0.147i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.28 - 3.94i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.104 + 0.104i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.14 + 6.60i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (7.33 - 5.32i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.77 + 3.44i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.27 + 1.06i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.91 - 3.91i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.479 + 0.942i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.644 - 4.07i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (6.16 + 2.00i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.59 + 7.70i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.94 - 2.94i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.02 + 0.744i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.57 + 0.804i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (3.51 - 2.55i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.29 - 8.18i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 4.23iT - 89T^{2} \) |
| 97 | \( 1 + (14.2 + 2.25i)T + (92.2 + 29.9i)T^{2} \) |
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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
−14.64566227065278130015479430956, −13.83680979591742069919042238125, −12.47291307099731678451487328508, −11.45817327028968632473177475602, −10.80095917392022751710144895544, −9.472083119655407697808742825496, −8.262863670904750974314420696131, −5.82710113884338191409097311365, −3.75540393706118718550007109149, −2.59701520720764262249289294467,
4.43156324889834294684685829463, 5.45949599845677075093769183312, 7.30055548764111973125705072783, 8.031710311998126932851293991139, 9.059505488153766098739144519256, 11.01435541906805374491078194338, 12.99577330202450036132298605371, 13.53552501459900913099251523875, 14.62279634542479032765775500110, 15.74388447147351289707746773436
