L(s) = 1 | + 1.34·2-s + (−1.11 − 1.32i)3-s − 0.184·4-s + (−1.26 − 2.19i)5-s + (−1.5 − 1.78i)6-s − 2.94·8-s + (−0.520 + 2.95i)9-s + (−1.70 − 2.95i)10-s + (−0.233 + 0.405i)11-s + (0.205 + 0.245i)12-s + (−2.91 + 5.04i)13-s + (−1.49 + 4.12i)15-s − 3.59·16-s + (−1.93 − 3.35i)17-s + (−0.701 + 3.98i)18-s + (1.09 − 1.89i)19-s + ⋯ |
L(s) = 1 | + 0.952·2-s + (−0.642 − 0.766i)3-s − 0.0923·4-s + (−0.566 − 0.980i)5-s + (−0.612 − 0.729i)6-s − 1.04·8-s + (−0.173 + 0.984i)9-s + (−0.539 − 0.934i)10-s + (−0.0705 + 0.122i)11-s + (0.0593 + 0.0707i)12-s + (−0.807 + 1.39i)13-s + (−0.387 + 1.06i)15-s − 0.899·16-s + (−0.470 − 0.814i)17-s + (−0.165 + 0.938i)18-s + (0.250 − 0.434i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0325542 + 0.512986i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0325542 + 0.512986i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.11 + 1.32i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + (1.26 + 2.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.233 - 0.405i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.91 - 5.04i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.93 + 3.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 1.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0530 - 0.0918i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + (-3.84 + 6.65i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + (-0.358 - 0.620i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 0.736T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + (-5.13 - 8.89i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + (-1.36 - 2.36i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.05 + 7.02i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.80 - 11.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.31212711429160931976910102233, −9.553861203865325146872309958852, −8.881917524908578160484195645292, −7.66181958604686024202201860194, −6.80262262145254060437597972153, −5.62594004901320904889964360985, −4.80731246549708467716868329916, −4.12173679141026505828035303466, −2.26320601607568600418535310076, −0.24802698083798666870835540963,
3.09697582623663454091332160905, 3.68941037594969802968216591028, 4.89114962578282056372463793187, 5.65399834147712095373816727147, 6.62337588031373120156903213480, 7.78932261266606445709485794385, 9.034530174418036862132225751711, 10.07895237226985240375452593975, 10.82632602378828751615507395636, 11.54426440876305501001428864531