L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + 0.999·20-s + (−0.499 + 0.866i)25-s − 0.999·27-s − 0.999·36-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s − 0.999·48-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + 0.999·20-s + (−0.499 + 0.866i)25-s − 0.999·27-s − 0.999·36-s + (0.499 − 0.866i)45-s + (−1 − 1.73i)47-s − 0.999·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.267208684\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.267208684\) |
\(L(1)\) |
|
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 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 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
−10.52637397281877366073093152534, −9.689503881927085556120129940704, −8.742794392041753860315530280391, −7.75555311219985220427541705270, −6.64341832279841336983118199946, −6.43537031924491084636755520879, −5.40760113346433075488120997423, −3.71211874767177296529468498933, −2.43476684438956917686842830607, −1.68298445597528661425097146136,
2.17800736411101080810521990340, 3.13171993246734502106979498647, 4.37078128411502001449074164289, 5.02964532871386960084274842880, 6.30718773024176410848288525851, 7.45699921777825010921990131947, 8.282295216607599097573046521155, 9.108103775034608040782763722498, 9.571897958238636919640746391690, 10.73943095168790138141127992420