| 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 | \( 1 + (0.988 + 1.71i)T + (-48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.12833911233876400468668473974, −10.34719020902677553246655171040, −9.592245156330890589723629218266, −8.696678027345445577296241551545, −6.57221272683358199070302318828, −6.06000920365093811909165234184, −4.95488267583204895522280958967, −4.38360228012602121572169615720, −3.19012892023972271529843964325, −1.62321262091518584877767990361,
2.01903530718103456626289021666, 3.06841639296105091628969433641, 5.10169698133639215504481650925, 5.56513386931934216828855285474, 6.22136292533911664855333563898, 6.99351917329554190624051943179, 8.080726910695958763905509097409, 9.676092709719828282714108286884, 10.23997950775507862501076027590, 11.69766640158316471608212529283
