L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.613·5-s + 0.999·8-s + (−0.306 − 0.531i)10-s + 4.62·11-s + (3.25 + 5.64i)13-s + (−0.5 − 0.866i)16-s + (1.01 + 1.75i)17-s + (1.32 − 2.29i)19-s + (−0.306 + 0.531i)20-s + (−2.31 − 4.00i)22-s + 3.62·23-s − 4.62·25-s + (3.25 − 5.64i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.274·5-s + 0.353·8-s + (−0.0970 − 0.168i)10-s + 1.39·11-s + (0.903 + 1.56i)13-s + (−0.125 − 0.216i)16-s + (0.246 + 0.426i)17-s + (0.303 − 0.525i)19-s + (−0.0686 + 0.118i)20-s + (−0.492 − 0.853i)22-s + 0.755·23-s − 0.924·25-s + (0.639 − 1.10i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.663827267\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.663827267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.613T + 5T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + (-3.25 - 5.64i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.01 - 1.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.32 + 2.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.25 - 5.64i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 1.75i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.31 + 2.27i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.29 + 9.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.11 - 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.56 - 6.17i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.31 - 4.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.376T + 71T^{2} \) |
| 73 | \( 1 + (-3.66 - 6.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.81 + 11.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.87 - 6.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.13 + 12.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.14 - 14.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.835532413373746200436300458040, −8.665258480221388222701796589155, −7.26326550969330322831295697319, −6.74839934621842932296672776242, −5.92311261036561230000277303813, −4.79461261030937088916512874856, −3.92294298912513117365102672706, −3.29451252065673083100598323492, −1.85759244927849902892385370747, −1.28896361913515283087969458896,
0.69154218078514815447702491241, 1.76157337641372659656382962143, 3.24382434724828319399101704310, 3.99039225497066192826135998347, 5.14334976979445399452539260264, 5.95765114468880912608899704583, 6.32441440680038382479045725561, 7.50351939613765285588266846684, 7.893611760253580547978444135613, 8.848665397805657786736737690634