| L(s) = 1 | − 1.41·2-s + (−2.92 − 0.650i)3-s + 2.00·4-s + (4.14 + 0.920i)6-s − 2.64i·7-s − 2.82·8-s + (8.15 + 3.81i)9-s − 7.97i·11-s + (−5.85 − 1.30i)12-s + 5.79i·13-s + 3.74i·14-s + 4.00·16-s + 9.35·17-s + (−11.5 − 5.39i)18-s + 22.7·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.976 − 0.216i)3-s + 0.500·4-s + (0.690 + 0.153i)6-s − 0.377i·7-s − 0.353·8-s + (0.905 + 0.423i)9-s − 0.725i·11-s + (−0.488 − 0.108i)12-s + 0.445i·13-s + 0.267i·14-s + 0.250·16-s + 0.550·17-s + (−0.640 − 0.299i)18-s + 1.19·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8846851050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8846851050\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (2.92 + 0.650i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 7.97iT - 121T^{2} \) |
| 13 | \( 1 - 5.79iT - 169T^{2} \) |
| 17 | \( 1 - 9.35T + 289T^{2} \) |
| 19 | \( 1 - 22.7T + 361T^{2} \) |
| 23 | \( 1 + 34.9T + 529T^{2} \) |
| 29 | \( 1 - 15.5iT - 841T^{2} \) |
| 31 | \( 1 - 25.5T + 961T^{2} \) |
| 37 | \( 1 - 38.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 1.03T + 2.20e3T^{2} \) |
| 53 | \( 1 - 41.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 77.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 33.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 54.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 78.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 24.5iT - 9.40e3T^{2} \) |
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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
−9.954308184847109033119444810375, −8.792723627570799042520029783241, −7.85431207969432316013127545930, −7.19354276038319661386809658012, −6.26356710484744518169659200982, −5.59263263851020827029734274184, −4.45923361478502901234559342766, −3.22766472889275854815295711866, −1.65337729740313547937314822031, −0.61852021209287948879499476596,
0.77302397203165687620582543078, 2.09035226254162821953404030630, 3.54440196816426509189954131387, 4.75112171213978489233893576170, 5.68311644595398809431846531579, 6.35384190186801812679337916381, 7.44544603182914967254684176791, 7.988744627422881079448997536578, 9.300447017296723860442585396682, 9.832457479641304930338222303802
