L(s) = 1 | + (2.15 − 1.24i)5-s + 0.872i·7-s − 2.41·11-s + (−1.98 + 3.43i)13-s + (−0.303 + 0.174i)17-s + (−1.35 − 4.14i)19-s + (3.88 − 6.72i)23-s + (0.608 − 1.05i)25-s + (7.96 + 4.59i)29-s − 2.43i·31-s + (1.08 + 1.88i)35-s + 6.18·37-s + (1.31 − 0.760i)41-s + (5.24 − 3.02i)43-s + (4.05 − 7.01i)47-s + ⋯ |
L(s) = 1 | + (0.965 − 0.557i)5-s + 0.329i·7-s − 0.726·11-s + (−0.549 + 0.951i)13-s + (−0.0734 + 0.0424i)17-s + (−0.311 − 0.950i)19-s + (0.809 − 1.40i)23-s + (0.121 − 0.210i)25-s + (1.47 + 0.853i)29-s − 0.437i·31-s + (0.183 + 0.318i)35-s + 1.01·37-s + (0.205 − 0.118i)41-s + (0.800 − 0.462i)43-s + (0.590 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.061350711\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.061350711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.35 + 4.14i)T \) |
good | 5 | \( 1 + (-2.15 + 1.24i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 0.872iT - 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + (1.98 - 3.43i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.303 - 0.174i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.88 + 6.72i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.96 - 4.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.43iT - 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + (-1.31 + 0.760i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.24 + 3.02i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.05 + 7.01i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.41 - 1.97i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.55 + 2.68i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.53 + 11.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.04 - 2.91i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.65 - 8.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.01 + 1.75i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (10.8 - 6.25i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.29T + 83T^{2} \) |
| 89 | \( 1 + (-14.0 - 8.11i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.90 + 15.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.896583489404880453310734703130, −8.207086168778919211383287148667, −7.05178782823304235492594852548, −6.54699038694214773429158195783, −5.53932399667637856769269936449, −4.95017360110416033301959315756, −4.22228701525201147759536307537, −2.65774691198408620565627353748, −2.22578122190804240586522574209, −0.798884709715659404816518305255,
1.02922042988157656198499392129, 2.39243139293711235451067256822, 2.95080748861040985524468526481, 4.12794364070320065677444221649, 5.17027091209629535576853936127, 5.82142436242162487002337691911, 6.48604324197262649389066735276, 7.53848212584913335869533309151, 7.890608351934975375560896067088, 8.989829984927461124837480135750