| L(s) = 1 | + (1.01 − 1.01i)2-s + (−1.22 − 1.22i)3-s + 1.94i·4-s − 2.48·6-s + (−1.87 + 1.87i)7-s + (6.02 + 6.02i)8-s + 2.99i·9-s − 10.7·11-s + (2.38 − 2.38i)12-s + (−15.5 − 15.5i)13-s + 3.78i·14-s + 4.41·16-s + (13.8 − 13.8i)17-s + (3.03 + 3.03i)18-s − 33.1i·19-s + ⋯ |
| L(s) = 1 | + (0.506 − 0.506i)2-s + (−0.408 − 0.408i)3-s + 0.487i·4-s − 0.413·6-s + (−0.267 + 0.267i)7-s + (0.753 + 0.753i)8-s + 0.333i·9-s − 0.973·11-s + (0.198 − 0.198i)12-s + (−1.19 − 1.19i)13-s + 0.270i·14-s + 0.275·16-s + (0.816 − 0.816i)17-s + (0.168 + 0.168i)18-s − 1.74i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9671444417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9671444417\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
| good | 2 | \( 1 + (-1.01 + 1.01i)T - 4iT^{2} \) |
| 11 | \( 1 + 10.7T + 121T^{2} \) |
| 13 | \( 1 + (15.5 + 15.5i)T + 169iT^{2} \) |
| 17 | \( 1 + (-13.8 + 13.8i)T - 289iT^{2} \) |
| 19 | \( 1 + 33.1iT - 361T^{2} \) |
| 23 | \( 1 + (8.39 + 8.39i)T + 529iT^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 - 44.7T + 961T^{2} \) |
| 37 | \( 1 + (21.6 - 21.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 22.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (9.41 + 9.41i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (46.0 - 46.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (51.4 + 51.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 5.55iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 49.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-38.8 + 38.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 23.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-60.1 - 60.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 8.04iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-5.76 - 5.76i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (21.9 - 21.9i)T - 9.40e3iT^{2} \) |
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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
−10.42703102628688299646075422829, −9.665250133417214372813314306412, −8.161625096235186750346504697903, −7.70726529654709317596357304497, −6.63503585301120676549785554735, −5.22747755188092831914042472654, −4.77515915682067742632905064650, −3.00715314795753048877923759504, −2.49721078017976630159688573389, −0.32492973966115146944750787318,
1.65663091430048088265070841326, 3.53907870296502223890874052369, 4.62255577258470817336130504557, 5.39995223377112013908667168385, 6.26951322311531892221029987490, 7.17246804821726502164446949646, 8.143249430440427549282617214204, 9.624593985967658313487538454569, 10.11706517084226445478713920121, 10.75954597789925157563015972922
