| L(s) = 1 | − 1.83i·3-s − 3.97i·7-s − 0.384·9-s + 2.10·11-s − 5.35i·13-s + 1.29i·17-s − 2.10·19-s − 7.30·21-s + i·23-s − 4.81i·27-s − 6.03·29-s − 8.32·31-s − 3.86i·33-s + 5.10i·37-s − 9.85·39-s + ⋯ |
| L(s) = 1 | − 1.06i·3-s − 1.50i·7-s − 0.128·9-s + 0.634·11-s − 1.48i·13-s + 0.314i·17-s − 0.482·19-s − 1.59·21-s + 0.208i·23-s − 0.926i·27-s − 1.11·29-s − 1.49·31-s − 0.673i·33-s + 0.838i·37-s − 1.57·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.320722590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.320722590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.83iT - 3T^{2} \) |
| 7 | \( 1 + 3.97iT - 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 + 5.35iT - 13T^{2} \) |
| 17 | \( 1 - 1.29iT - 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 - 5.10iT - 37T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 + 7.78iT - 43T^{2} \) |
| 47 | \( 1 - 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 0.573iT - 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 8.36iT - 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 1.51iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.337iT - 97T^{2} \) |
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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
−7.85847184283244990664206651480, −7.11381192001301216508203816050, −6.73913473956267232942444178746, −5.85935980152266814740372095800, −4.99634076475822456263783545148, −3.89752492065479780799827205406, −3.45724220580118112726268976620, −2.08001246605113863262942439326, −1.23794814030020725084723139214, −0.36652519075668368638498649860,
1.74025146257455691680137716623, 2.42980949879126577265849039031, 3.75330219561334956552961528983, 4.04976630355794678360196001531, 5.21435530156251372997691965000, 5.47956076951387378961178767324, 6.63493736029612104557260005638, 7.05875340927609244250215467719, 8.392777450473058697972825351144, 8.850824284931532181543344861267
