L(s) = 1 | + (−0.499 − 1.32i)2-s + (−2.08 − 1.20i)3-s + (−1.50 + 1.32i)4-s − 2.36·5-s + (−0.551 + 3.35i)6-s + (1.42 + 2.46i)7-s + (2.49 + 1.32i)8-s + (1.39 + 2.40i)9-s + (1.17 + 3.12i)10-s + (3.21 + 0.819i)11-s + (4.71 − 0.946i)12-s + (−2.34 − 2.74i)13-s + (2.54 − 3.11i)14-s + (4.92 + 2.84i)15-s + (0.508 − 3.96i)16-s + (−1.53 + 0.886i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.935i)2-s + (−1.20 − 0.694i)3-s + (−0.750 + 0.660i)4-s − 1.05·5-s + (−0.224 + 1.36i)6-s + (0.537 + 0.931i)7-s + (0.883 + 0.469i)8-s + (0.463 + 0.802i)9-s + (0.373 + 0.988i)10-s + (0.968 + 0.247i)11-s + (1.36 − 0.273i)12-s + (−0.649 − 0.760i)13-s + (0.681 − 0.831i)14-s + (1.27 + 0.733i)15-s + (0.127 − 0.991i)16-s + (−0.372 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.371269 - 0.426004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.371269 - 0.426004i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.499 + 1.32i)T \) |
| 11 | \( 1 + (-3.21 - 0.819i)T \) |
| 13 | \( 1 + (2.34 + 2.74i)T \) |
good | 3 | \( 1 + (2.08 + 1.20i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 + (-1.42 - 2.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 17 | \( 1 + (1.53 - 0.886i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0598 + 0.103i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.93 - 3.42i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.88 + 1.66i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.06iT - 31T^{2} \) |
| 37 | \( 1 + (-2.26 + 3.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.34 - 4.23i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.01 + 6.94i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.87iT - 47T^{2} \) |
| 53 | \( 1 - 5.47T + 53T^{2} \) |
| 59 | \( 1 + (-12.7 + 7.36i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.95 + 1.13i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.27 - 4.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.24 - 0.716i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 15.8iT - 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 + (-3.41 + 5.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.294 + 0.510i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.04243016393132669057579496053, −9.703180145439699930233356460926, −8.797620936148685602614129673483, −7.81027885760267916987212417604, −7.11576269627982218875822760672, −5.75020896126781884464810386069, −4.84594671916712387098681497602, −3.69707962483856668156104809582, −2.14282963236203785224656198528, −0.66259443724711326550250692261,
0.856287190089151949197204663140, 4.01311738177327962356348296189, 4.48793224990731049893717509785, 5.35861004284051005296814925993, 6.70495136069604707713143359479, 7.11413234551084162733181447205, 8.245042503446753635549515287558, 9.183817311185948764821194056242, 10.14209677038719911972924022308, 11.05193194043867422875925377916