L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 1.41·6-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (−0.5 + 0.866i)12-s − 0.999·15-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)18-s + (0.707 − 1.22i)19-s − 0.999·20-s + (0.707 − 1.22i)23-s + (−0.499 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 1.41·6-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (−0.5 + 0.866i)12-s − 0.999·15-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)18-s + (0.707 − 1.22i)19-s − 0.999·20-s + (0.707 − 1.22i)23-s + (−0.499 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5598950061\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5598950061\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.34993120347775122554551043102, −9.218322757749690495085919282960, −8.767564308051989910992406233728, −7.80872387016918414946977913691, −7.12523073467722465423152838236, −6.27063349225231279058608907105, −5.54555463523792179981987746177, −4.70840255113185615258939622988, −2.54939778975243280944115154810, −0.857268764279335774323015240523,
1.63469389138249823907523463923, 3.10681241598429729348555652662, 3.65925848659690912855683389582, 5.26355169963314242333493287683, 6.05526125403350144727956635115, 7.19367491245394962018639464830, 8.495377593257588340567644740661, 9.494713869916197309669131178067, 9.849371985023292272587692959458, 10.64378474079581606898307340882