| L(s) = 1 | + (−2.04 + 2.04i)3-s + (0.701 + 0.701i)5-s + i·7-s − 5.33i·9-s + (2.41 + 2.41i)11-s + (1.96 − 1.96i)13-s − 2.86·15-s − 6.93·17-s + (−1.38 + 1.38i)19-s + (−2.04 − 2.04i)21-s + 2.05i·23-s − 4.01i·25-s + (4.76 + 4.76i)27-s + (−5.34 + 5.34i)29-s − 5.23·31-s + ⋯ |
| L(s) = 1 | + (−1.17 + 1.17i)3-s + (0.313 + 0.313i)5-s + 0.377i·7-s − 1.77i·9-s + (0.729 + 0.729i)11-s + (0.543 − 0.543i)13-s − 0.739·15-s − 1.68·17-s + (−0.318 + 0.318i)19-s + (−0.445 − 0.445i)21-s + 0.428i·23-s − 0.803i·25-s + (0.917 + 0.917i)27-s + (−0.992 + 0.992i)29-s − 0.940·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3250667300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3250667300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (2.04 - 2.04i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.701 - 0.701i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.41 - 2.41i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.96 + 1.96i)T - 13iT^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + (1.38 - 1.38i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.05iT - 23T^{2} \) |
| 29 | \( 1 + (5.34 - 5.34i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + (-6.58 - 6.58i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.949iT - 41T^{2} \) |
| 43 | \( 1 + (-5.95 - 5.95i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + (7.24 + 7.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (8.58 + 8.58i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.81 - 2.81i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.07 - 9.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.60iT - 71T^{2} \) |
| 73 | \( 1 + 6.53iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + (-4.49 + 4.49i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.428iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.770739811696654549543816510412, −9.312152114849385996280723793199, −8.459587417665652378017023236239, −7.15896767768723944946770630549, −6.26080450741015503129370210429, −5.88526364146655693740679684544, −4.77790120027319991983675228906, −4.28131913194922280767936372743, −3.21786497157372975593323087637, −1.75947007409388953384137122978,
0.14561565206129270595639391822, 1.31832173384720038783254967321, 2.21403381626498920251205253856, 3.90386634428660450310170794085, 4.76281649387903379363771255322, 6.00217098288716056787940239655, 6.14806392234376845279174224268, 7.06983927011621225376695337971, 7.70443507883953377664617361762, 8.918552471329088699337581312844
