L(s) = 1 | + (1 + 1.41i)3-s + (−1 + 2i)5-s + (−0.414 + 0.414i)7-s + (−1.00 + 2.82i)9-s + 4.82i·11-s + (1.82 + 1.82i)13-s + (−3.82 + 0.585i)15-s + (−3.82 − 3.82i)17-s − 4.82i·19-s + (−1 − 0.171i)21-s + (1.58 − 1.58i)23-s + (−3 − 4i)25-s + (−5.00 + 1.41i)27-s − 7.65·29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s + (−0.447 + 0.894i)5-s + (−0.156 + 0.156i)7-s + (−0.333 + 0.942i)9-s + 1.45i·11-s + (0.507 + 0.507i)13-s + (−0.988 + 0.151i)15-s + (−0.928 − 0.928i)17-s − 1.10i·19-s + (−0.218 − 0.0374i)21-s + (0.330 − 0.330i)23-s + (−0.600 − 0.800i)25-s + (−0.962 + 0.272i)27-s − 1.42·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.270343 + 1.32770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270343 + 1.32770i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 + (1 - 2i)T \) |
good | 7 | \( 1 + (0.414 - 0.414i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (-1.82 - 1.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.82 + 3.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.82iT - 19T^{2} \) |
| 23 | \( 1 + (-1.58 + 1.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2.41 - 2.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.41 - 6.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (3 - 3i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + (4.07 - 4.07i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.48iT - 71T^{2} \) |
| 73 | \( 1 + (-6.65 - 6.65i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.82iT - 79T^{2} \) |
| 83 | \( 1 + (-5.24 + 5.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−10.37815404907988006199021934646, −9.387935628714755275846288653118, −9.071212114220041475750587517811, −7.77228214935139162860620999426, −7.15803563621419510156831220398, −6.24890071207497508310081361319, −4.73379433017198239157751104658, −4.28697300955504116747875964274, −3.01701986230742552138794534244, −2.25912475447771657539560521372,
0.57596062357488966673279131095, 1.80823860558894154241631116507, 3.37683203393207028424491795026, 3.96084043418987948536286852103, 5.55666905155405264326818837028, 6.15709844446410012454275722110, 7.30627065845049420795337535076, 8.250397818680708701912821622962, 8.533191544568383549255297825888, 9.346276431365443806666630208016