L(s) = 1 | + 1.73·2-s − 1.73·3-s + 0.999·4-s − 2.99·6-s + (−1.73 − 2i)7-s − 1.73·8-s + 2.99·9-s − 3.46i·11-s − 1.73·12-s + (−2.99 − 3.46i)14-s − 5·16-s − 6i·17-s + 5.19·18-s − 3.46i·19-s + (2.99 + 3.46i)21-s − 5.99i·22-s + ⋯ |
L(s) = 1 | + 1.22·2-s − 1.00·3-s + 0.499·4-s − 1.22·6-s + (−0.654 − 0.755i)7-s − 0.612·8-s + 0.999·9-s − 1.04i·11-s − 0.500·12-s + (−0.801 − 0.925i)14-s − 1.25·16-s − 1.45i·17-s + 1.22·18-s − 0.794i·19-s + (0.654 + 0.755i)21-s − 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.647910 - 0.970428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.647910 - 0.970428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−10.98490035414680524046213969541, −9.869296639066506589817132618294, −8.995143056474498140871021607547, −7.40200064427401613730141765913, −6.59457783530331046362937615326, −5.78910825759643750426958902774, −4.92315598764960746724324685214, −4.00272055187375551796376162852, −2.95354040378269973284502007338, −0.50845754913181087462085742301,
2.13418116898562460899790806945, 3.75202542170310736445502216036, 4.50497036570890397574330743705, 5.70811871399084912584138353943, 6.05395370068521863919986986343, 7.06236171267271289365881457155, 8.432087354694194374882295233620, 9.697293307356519553725721407905, 10.27961195982735322551314500871, 11.58807786512852628901509660164