| L(s) = 1 | + (1.03 + 0.964i)2-s + (0.511 − 0.704i)3-s + (0.138 + 1.99i)4-s + (1.92 + 1.13i)5-s + (1.20 − 0.234i)6-s + (−1.80 + 1.80i)7-s + (−1.78 + 2.19i)8-s + (0.692 + 2.13i)9-s + (0.898 + 3.03i)10-s + (−2.70 − 5.31i)11-s + (1.47 + 0.923i)12-s + (1.67 + 5.16i)13-s + (−3.60 + 0.124i)14-s + (1.78 − 0.777i)15-s + (−3.96 + 0.552i)16-s + (1.24 − 7.85i)17-s + ⋯ |
| L(s) = 1 | + (0.731 + 0.682i)2-s + (0.295 − 0.406i)3-s + (0.0692 + 0.997i)4-s + (0.861 + 0.507i)5-s + (0.493 − 0.0957i)6-s + (−0.680 + 0.680i)7-s + (−0.629 + 0.776i)8-s + (0.230 + 0.710i)9-s + (0.284 + 0.958i)10-s + (−0.816 − 1.60i)11-s + (0.426 + 0.266i)12-s + (0.465 + 1.43i)13-s + (−0.962 + 0.0333i)14-s + (0.460 − 0.200i)15-s + (−0.990 + 0.138i)16-s + (0.301 − 1.90i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70676 + 1.50678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70676 + 1.50678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.03 - 0.964i)T \) |
| 5 | \( 1 + (-1.92 - 1.13i)T \) |
| good | 3 | \( 1 + (-0.511 + 0.704i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (1.80 - 1.80i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.70 + 5.31i)T + (-6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-1.67 - 5.16i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.24 + 7.85i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.137 - 0.867i)T + (-18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (-5.79 + 2.95i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (0.276 + 1.74i)T + (-27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (0.136 + 0.187i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.42 + 4.39i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.96 + 0.638i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 + (-0.0310 - 0.195i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.65 + 2.28i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.95 - 5.79i)T + (-34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (3.16 + 6.21i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-4.80 + 3.48i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.474 - 0.344i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.33 + 8.50i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (3.06 + 2.22i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (8.80 + 12.1i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (4.38 - 13.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.08 + 0.646i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.55418653464880995982277088104, −10.79750317556680931493168440792, −9.319698637849624579893892629366, −8.713483995528018591061135963912, −7.49040334774666578738782003169, −6.66212443703166865026058040411, −5.82798651418614556150812562503, −4.94656600449028854854948947463, −3.14909123887893546641058743275, −2.45802470214295911362862808279,
1.34457482894191540314292378537, 2.94127517929872894196939412358, 3.97084468257016379780825471779, 5.05263133701456245318595417485, 6.01723141960449446022939427996, 7.07635378402032586294075560892, 8.591581277191151082787226269255, 9.784631909157381771961199772545, 10.09500048930428080914988256714, 10.75445934383971244538588806886
