| L(s) = 1 | + (1.87 + 0.903i)2-s + (−1.02 − 1.39i)3-s + (1.45 + 1.82i)4-s + (3.54 + 1.09i)5-s + (−0.654 − 3.54i)6-s + (−0.389 + 2.61i)7-s + (0.154 + 0.678i)8-s + (−0.909 + 2.85i)9-s + (5.66 + 5.25i)10-s + (0.165 + 2.21i)11-s + (1.06 − 3.90i)12-s + (−0.363 − 4.85i)13-s + (−3.09 + 4.55i)14-s + (−2.09 − 6.07i)15-s + (0.716 − 3.13i)16-s + (−1.40 − 3.56i)17-s + ⋯ |
| L(s) = 1 | + (1.32 + 0.638i)2-s + (−0.590 − 0.807i)3-s + (0.727 + 0.912i)4-s + (1.58 + 0.489i)5-s + (−0.267 − 1.44i)6-s + (−0.147 + 0.989i)7-s + (0.0547 + 0.240i)8-s + (−0.303 + 0.952i)9-s + (1.79 + 1.66i)10-s + (0.0500 + 0.667i)11-s + (0.307 − 1.12i)12-s + (−0.100 − 1.34i)13-s + (−0.826 + 1.21i)14-s + (−0.541 − 1.56i)15-s + (0.179 − 0.784i)16-s + (−0.339 − 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.64007 + 0.864128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.64007 + 0.864128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.02 + 1.39i)T \) |
| 7 | \( 1 + (0.389 - 2.61i)T \) |
| good | 2 | \( 1 + (-1.87 - 0.903i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-3.54 - 1.09i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.165 - 2.21i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (0.363 + 4.85i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (1.40 + 3.56i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.03 - 5.25i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.24 - 0.640i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (0.721 + 1.83i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 + (3.95 + 0.595i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (0.404 - 0.375i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (9.53 + 8.84i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (0.548 + 0.264i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (8.57 - 1.29i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-1.93 + 8.49i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.249 + 0.313i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + (-10.0 - 12.6i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.134 + 1.78i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 + (0.862 - 11.5i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.636 + 0.433i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (0.988 - 1.71i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.69766640158316471608212529283, −10.23997950775507862501076027590, −9.676092709719828282714108286884, −8.080726910695958763905509097409, −6.99351917329554190624051943179, −6.22136292533911664855333563898, −5.56513386931934216828855285474, −5.10169698133639215504481650925, −3.06841639296105091628969433641, −2.01903530718103456626289021666,
1.62321262091518584877767990361, 3.19012892023972271529843964325, 4.38360228012602121572169615720, 4.95488267583204895522280958967, 6.06000920365093811909165234184, 6.57221272683358199070302318828, 8.696678027345445577296241551545, 9.592245156330890589723629218266, 10.34719020902677553246655171040, 11.12833911233876400468668473974
