L(s) = 1 | + (−0.5 + 0.866i)2-s − i·3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (−0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + (−0.866 + 1.5i)17-s + (0.5 − 0.866i)18-s + (0.866 + 1.5i)19-s + (−0.5 + 0.866i)22-s − 0.999i·24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s − i·3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (−0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + (−0.866 + 1.5i)17-s + (0.5 − 0.866i)18-s + (0.866 + 1.5i)19-s + (−0.5 + 0.866i)22-s − 0.999i·24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9020106957\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9020106957\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.473055568583809591988491708163, −8.250590609868790518838887343899, −7.27165743128968067827041975458, −6.68957645211901499637534085912, −6.10871872416341367082888780216, −5.46665749277556177612323546130, −4.35952299403539024848344469871, −3.40396233842231186169105520037, −1.88463744275839813241971611435, −1.21149292652454301434211549745,
0.73147836974766753300425314296, 2.30921815260248950664228548187, 3.08916462947807128443092321381, 3.84180064589181188000014588392, 4.81662428547783065456760174437, 5.12304320534923562618298278107, 6.65703327617474012913310203958, 7.16621093452617221739547174683, 8.331307918414042740266182103164, 8.972550626167594234186939324881