L(s) = 1 | + 2-s + 4-s + (−0.5 − 2.59i)7-s + 8-s + (3 + 5.19i)11-s + (−2.5 − 4.33i)13-s + (−0.5 − 2.59i)14-s + 16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (3 + 5.19i)22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−2.5 − 4.33i)26-s + (−0.5 − 2.59i)28-s + (3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (0.904 + 1.56i)11-s + (−0.693 − 1.20i)13-s + (−0.133 − 0.694i)14-s + 0.250·16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (0.639 + 1.10i)22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.490 − 0.849i)26-s + (−0.0944 − 0.490i)28-s + (0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.613231022\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613231022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.07410273012661055759578126232, −9.079530922070143779683080894501, −7.53110057991743244949354691771, −7.40621920555462411158286841358, −6.44416954844267361020173181362, −5.24357758870821828163612566850, −4.58314249904603549018329730152, −3.61238424405171559155945006030, −2.58924677193309664956985900145, −1.05265602894310832411085821794,
1.45685937729851190905882712233, 2.84334288646061969186461333611, 3.61524966323110508303152505132, 4.76574011115947488832893584227, 5.69751939932530624951526541798, 6.35293121075353093682762361401, 7.15235118997075856624656702725, 8.436941407549846736404055352110, 8.959865939426136703937909115793, 9.834200577861963653558935819176