| L(s) = 1 | + (1.78 − 2.40i)3-s − 12.2·7-s + (−2.60 − 8.61i)9-s + 8.79i·11-s − 6.20·13-s − 16.6i·17-s + 8.93·19-s + (−21.8 + 29.3i)21-s + 40.1i·23-s + (−25.4 − 9.14i)27-s − 4.94i·29-s + 50.6·31-s + (21.1 + 15.7i)33-s + 41.3·37-s + (−11.0 + 14.9i)39-s + ⋯ |
| L(s) = 1 | + (0.596 − 0.802i)3-s − 1.74·7-s + (−0.288 − 0.957i)9-s + 0.799i·11-s − 0.477·13-s − 0.977i·17-s + 0.470·19-s + (−1.03 + 1.39i)21-s + 1.74i·23-s + (−0.940 − 0.338i)27-s − 0.170i·29-s + 1.63·31-s + (0.641 + 0.476i)33-s + 1.11·37-s + (−0.284 + 0.382i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.145289909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.145289909\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.78 + 2.40i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 12.2T + 49T^{2} \) |
| 11 | \( 1 - 8.79iT - 121T^{2} \) |
| 13 | \( 1 + 6.20T + 169T^{2} \) |
| 17 | \( 1 + 16.6iT - 289T^{2} \) |
| 19 | \( 1 - 8.93T + 361T^{2} \) |
| 23 | \( 1 - 40.1iT - 529T^{2} \) |
| 29 | \( 1 + 4.94iT - 841T^{2} \) |
| 31 | \( 1 - 50.6T + 961T^{2} \) |
| 37 | \( 1 - 41.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 78.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 96.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 34.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 30.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 108.T + 9.40e3T^{2} \) |
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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
−9.605162993843952353466235160490, −9.093833089200862038897434861917, −7.78255527792454950464386932423, −7.25930767227182074361091385955, −6.51909082191346974276021022585, −5.71608554590963350675721949292, −4.34675042257060402089847674594, −3.13596339796904305350085891965, −2.63426799135026232014781582760, −1.09072473077946496051665073269,
0.34580730465993342891469478509, 2.45188370680400912019002686023, 3.23669012095924691684943704699, 3.98124971631927938887090453905, 5.08868530392294843309736077060, 6.18619904691150288092528699203, 6.77067982790411528391821367007, 8.120635479715464181437102818306, 8.637405288185488616005633652855, 9.594252356444019356382624197203
