L(s) = 1 | + 2.17·2-s + (0.507 + 1.65i)3-s + 2.73·4-s + (0.634 + 1.09i)5-s + (1.10 + 3.60i)6-s + 1.60·8-s + (−2.48 + 1.68i)9-s + (1.38 + 2.39i)10-s + (2.73 − 4.74i)11-s + (1.38 + 4.53i)12-s + (−2.37 + 4.10i)13-s + (−1.49 + 1.60i)15-s − 1.98·16-s + (−2.40 − 4.17i)17-s + (−5.40 + 3.65i)18-s + (2.69 − 4.66i)19-s + ⋯ |
L(s) = 1 | + 1.53·2-s + (0.292 + 0.956i)3-s + 1.36·4-s + (0.283 + 0.491i)5-s + (0.450 + 1.47i)6-s + 0.566·8-s + (−0.828 + 0.560i)9-s + (0.436 + 0.755i)10-s + (0.825 − 1.43i)11-s + (0.400 + 1.30i)12-s + (−0.658 + 1.13i)13-s + (−0.386 + 0.415i)15-s − 0.496·16-s + (−0.584 − 1.01i)17-s + (−1.27 + 0.862i)18-s + (0.617 − 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.97032 + 1.56264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.97032 + 1.56264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.507 - 1.65i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 + (-0.634 - 1.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.73 + 4.74i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.37 - 4.10i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.40 + 4.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.58 - 4.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 3.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.94 + 3.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.66 + 2.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 + (-3.57 - 6.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 0.308T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 + (5.27 + 9.13i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + (-5.08 - 8.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.59 + 4.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.48 - 4.30i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.47595188595278163383981049390, −10.71740729398855112384599495440, −9.320576846968704796097106175323, −8.917193027885735125545541694808, −7.12620057726172638669333322719, −6.31542245312717411056977558948, −5.22324282490256167208409682399, −4.47825601529573405845478804134, −3.36821032406386108144992665661, −2.63209578258505084563569987606,
1.69352335455853798936813433246, 2.90244629536660429025497952339, 4.17867861927344967943343337080, 5.19587505183156716872319826193, 6.16281819848388798578864746602, 6.96370542689883943948476983040, 7.975139694061799152994128288499, 9.102627570522947512926691052181, 10.18336254890279388965960620890, 11.55984633716612968016762742867