L(s) = 1 | + (1 − i)3-s + (−1 + 2i)5-s + (3 + 3i)7-s + i·9-s + 2i·11-s + (−3 − 3i)13-s + (1 + 3i)15-s + (1 − i)17-s + 4·19-s + 6·21-s + (1 − i)23-s + (−3 − 4i)25-s + (4 + 4i)27-s − 10i·31-s + (2 + 2i)33-s + ⋯ |
L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (1.13 + 1.13i)7-s + 0.333i·9-s + 0.603i·11-s + (−0.832 − 0.832i)13-s + (0.258 + 0.774i)15-s + (0.242 − 0.242i)17-s + 0.917·19-s + 1.30·21-s + (0.208 − 0.208i)23-s + (−0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s − 1.79i·31-s + (0.348 + 0.348i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51698 + 0.430942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51698 + 0.430942i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + (-5 + 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (1 + i)T + 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−11.81591366705771694113443395954, −10.91406864582119726877925109974, −9.863288933263972862940294754371, −8.664646218729736647431524457543, −7.65693164585946708203716146088, −7.41312387154114080138329532482, −5.76657328954933454743612435381, −4.72433740835131608138449855835, −2.92838589410769449520653675639, −2.09253552181524448462198734667,
1.25787883574846595639059045570, 3.42519377028204504433773615563, 4.39210322439505830010682398136, 5.16562771064088594595218800447, 6.96600513175471902469013514923, 7.926287177670478020845990093964, 8.721526712404733872751331346150, 9.574229728358341599138416755870, 10.57734156767100509843644300402, 11.60867118319147795173080593192