L(s) = 1 | + 0.528i·3-s − 5-s + (2.17 + 1.50i)7-s + 2.72·9-s − 3.04·11-s + 4.75·13-s − 0.528i·15-s − 6.78i·17-s − 0.584i·19-s + (−0.796 + 1.14i)21-s + 5.80i·23-s + 25-s + 3.02i·27-s − 0.185i·29-s + 3.12·31-s + ⋯ |
L(s) = 1 | + 0.304i·3-s − 0.447·5-s + (0.821 + 0.569i)7-s + 0.907·9-s − 0.918·11-s + 1.31·13-s − 0.136i·15-s − 1.64i·17-s − 0.134i·19-s + (−0.173 + 0.250i)21-s + 1.21i·23-s + 0.200·25-s + 0.581i·27-s − 0.0344i·29-s + 0.561·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.775465228\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.775465228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.17 - 1.50i)T \) |
good | 3 | \( 1 - 0.528iT - 3T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 6.78iT - 17T^{2} \) |
| 19 | \( 1 + 0.584iT - 19T^{2} \) |
| 23 | \( 1 - 5.80iT - 23T^{2} \) |
| 29 | \( 1 + 0.185iT - 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 7.04iT - 37T^{2} \) |
| 41 | \( 1 - 3.83iT - 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 0.535iT - 53T^{2} \) |
| 59 | \( 1 + 1.52iT - 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 9.68iT - 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.81iT - 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 3.10iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−9.880985267396643682903096993220, −9.112329629488584282510519189213, −8.193421089185484136445985346046, −7.59832584813144202646297245149, −6.64624705957000998021518777728, −5.39426456290137485670123751556, −4.82989425260949892193967022025, −3.78416952095104137944463016504, −2.66012562213811317430690578927, −1.22155211821892572429915485596,
0.997187102909741545647440440454, 2.15254157518453180757076059161, 3.79983820942733046989609521615, 4.30577462474064340190220924867, 5.49924272848775298135025575574, 6.50667534777758761543438282688, 7.34332745365088446983894605181, 8.233092838422216166463235392539, 8.497462510347926420140944646563, 9.975358794216561766423322157353