| L(s) = 1 | + (0.454 + 1.15i)2-s + (1.71 + 0.239i)3-s + (0.333 − 0.309i)4-s + (−0.554 − 0.266i)5-s + (0.501 + 2.09i)6-s + (−1.17 + 2.37i)7-s + (2.74 + 1.32i)8-s + (2.88 + 0.821i)9-s + (0.0571 − 0.762i)10-s + (0.391 − 0.491i)11-s + (0.646 − 0.451i)12-s + (−0.676 − 1.72i)13-s + (−3.27 − 0.282i)14-s + (−0.886 − 0.590i)15-s + (−0.215 + 2.87i)16-s + (−0.396 − 0.368i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.818i)2-s + (0.990 + 0.138i)3-s + (0.166 − 0.154i)4-s + (−0.247 − 0.119i)5-s + (0.204 + 0.854i)6-s + (−0.444 + 0.895i)7-s + (0.972 + 0.468i)8-s + (0.961 + 0.273i)9-s + (0.0180 − 0.241i)10-s + (0.118 − 0.148i)11-s + (0.186 − 0.130i)12-s + (−0.187 − 0.477i)13-s + (−0.875 − 0.0756i)14-s + (−0.228 − 0.152i)15-s + (−0.0538 + 0.718i)16-s + (−0.0962 − 0.0893i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98644 + 1.31222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98644 + 1.31222i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 - 0.239i)T \) |
| 7 | \( 1 + (1.17 - 2.37i)T \) |
| good | 2 | \( 1 + (-0.454 - 1.15i)T + (-1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (0.554 + 0.266i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-0.391 + 0.491i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.676 + 1.72i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (0.396 + 0.368i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.957 + 1.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 6.10i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-3.40 + 1.05i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (2.65 + 4.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.58 - 1.41i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (0.614 - 0.418i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (2.00 + 1.36i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (0.325 + 0.828i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (3.90 + 1.20i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-1.09 - 0.748i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (7.80 + 7.24i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (3.06 + 5.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.464 - 2.03i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.428 - 0.0646i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-3.84 + 6.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.35 - 11.0i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-4.94 + 12.5i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-0.579 - 1.00i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.29579065093356730995005418751, −10.17380985909040033973165566135, −9.345847780711473612685253793521, −8.380627423919484751769870233200, −7.64288896287303285281526939495, −6.66826748965231471389265722847, −5.65955431038613485718617301210, −4.63317598333049860248753576349, −3.27913132731233968391792683713, −2.03064077083657076708960840501,
1.57138725562812197495629923124, 2.87703878681901034652603678373, 3.76480023758058948145972248324, 4.54379434199142941892459700756, 6.67147948355204448648581616092, 7.23855176141920402888870664760, 8.184675597646070183063092439401, 9.289299652088083345238868100693, 10.26990856272974984095666652561, 10.82377474958107642702553520163
