L(s) = 1 | + (−1.67 + 1.67i)2-s − 3-s − 3.58i·4-s + (2.12 + 2.12i)5-s + (1.67 − 1.67i)6-s + (−1.37 − 1.37i)7-s + (2.64 + 2.64i)8-s + 9-s − 7.10·10-s + (2.22 − 2.46i)11-s + 3.58i·12-s + (−0.554 + 3.56i)13-s + 4.58·14-s + (−2.12 − 2.12i)15-s − 1.68·16-s + 6.69·17-s + ⋯ |
L(s) = 1 | + (−1.18 + 1.18i)2-s − 0.577·3-s − 1.79i·4-s + (0.950 + 0.950i)5-s + (0.682 − 0.682i)6-s + (−0.518 − 0.518i)7-s + (0.936 + 0.936i)8-s + 0.333·9-s − 2.24·10-s + (0.669 − 0.742i)11-s + 1.03i·12-s + (−0.153 + 0.988i)13-s + 1.22·14-s + (−0.548 − 0.548i)15-s − 0.420·16-s + 1.62·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.400502 + 0.597093i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.400502 + 0.597093i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + (-2.22 + 2.46i)T \) |
| 13 | \( 1 + (0.554 - 3.56i)T \) |
good | 2 | \( 1 + (1.67 - 1.67i)T - 2iT^{2} \) |
| 5 | \( 1 + (-2.12 - 2.12i)T + 5iT^{2} \) |
| 7 | \( 1 + (1.37 + 1.37i)T + 7iT^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 + (-2.46 + 2.46i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.28iT - 23T^{2} \) |
| 29 | \( 1 - 3.36iT - 29T^{2} \) |
| 31 | \( 1 + (-0.273 - 0.273i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.74 - 5.74i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.977 - 0.977i)T - 41iT^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + (5.10 - 5.10i)T - 47iT^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 + (2.90 - 2.90i)T - 59iT^{2} \) |
| 61 | \( 1 - 6.86iT - 61T^{2} \) |
| 67 | \( 1 + (5.85 + 5.85i)T + 67iT^{2} \) |
| 71 | \( 1 + (-11.6 - 11.6i)T + 71iT^{2} \) |
| 73 | \( 1 + (-2.50 - 2.50i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.392iT - 79T^{2} \) |
| 83 | \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.52 - 5.52i)T - 89iT^{2} \) |
| 97 | \( 1 + (8.21 + 8.21i)T + 97iT^{2} \) |
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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
−11.06711303024636775892485942388, −10.09070038266560547583067817372, −9.738821734396592171322476567185, −8.805130423380231996173799251131, −7.45504757103582042779438157322, −6.81526685227295134520161974809, −6.19060911105199547738358676420, −5.34641796314851078783664116216, −3.37991494605112863648924341490, −1.25329216584913031266242097081,
0.892568324736893238073973323327, 2.05047489527011647113449972127, 3.49185016790531365725859813535, 5.18676869830693993015492013820, 5.99262545043996642607899464677, 7.52271551471256260461014807061, 8.535608699122837196388633437032, 9.517601453132148636378223633535, 9.841410844373535405817906101801, 10.58386532527453166559290619857