L(s) = 1 | + (0.893 + 0.448i)2-s + (−0.686 − 0.727i)3-s + (0.597 + 0.802i)4-s + (−0.286 − 0.957i)6-s + (0.173 + 0.984i)8-s + (−0.0581 + 0.998i)9-s + (1.16 − 0.275i)11-s + (0.173 − 0.984i)12-s + (−0.286 + 0.957i)16-s + (1.57 − 0.571i)17-s + (−0.500 + 0.866i)18-s + (−1.82 − 0.665i)19-s + (1.16 + 0.275i)22-s + (0.597 − 0.802i)24-s + (−0.835 + 0.549i)25-s + ⋯ |
L(s) = 1 | + (0.893 + 0.448i)2-s + (−0.686 − 0.727i)3-s + (0.597 + 0.802i)4-s + (−0.286 − 0.957i)6-s + (0.173 + 0.984i)8-s + (−0.0581 + 0.998i)9-s + (1.16 − 0.275i)11-s + (0.173 − 0.984i)12-s + (−0.286 + 0.957i)16-s + (1.57 − 0.571i)17-s + (−0.500 + 0.866i)18-s + (−1.82 − 0.665i)19-s + (1.16 + 0.275i)22-s + (0.597 − 0.802i)24-s + (−0.835 + 0.549i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.291380429\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.291380429\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.893 - 0.448i)T \) |
| 3 | \( 1 + (0.686 + 0.727i)T \) |
good | 5 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 7 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 11 | \( 1 + (-1.16 + 0.275i)T + (0.893 - 0.448i)T^{2} \) |
| 13 | \( 1 + (-0.396 - 0.918i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 0.571i)T + (0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (1.82 + 0.665i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.686 + 0.727i)T^{2} \) |
| 29 | \( 1 + (0.993 - 0.116i)T^{2} \) |
| 31 | \( 1 + (-0.973 + 0.230i)T^{2} \) |
| 37 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 41 | \( 1 + (1.22 - 0.615i)T + (0.597 - 0.802i)T^{2} \) |
| 43 | \( 1 + (1.22 + 1.30i)T + (-0.0581 + 0.998i)T^{2} \) |
| 47 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.558 + 0.132i)T + (0.893 + 0.448i)T^{2} \) |
| 61 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 67 | \( 1 + (0.0460 - 0.790i)T + (-0.993 - 0.116i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (0.0201 + 0.114i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 83 | \( 1 + (-1.36 - 0.687i)T + (0.597 + 0.802i)T^{2} \) |
| 89 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.569 + 1.90i)T + (-0.835 - 0.549i)T^{2} \) |
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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
−11.24488381351012770589532151385, −10.16848579141557827292856247454, −8.782673305750210132345062960983, −7.900944363436951773435078820956, −6.97538724980102166591150841236, −6.33503169881069216898373759701, −5.50090746785284466708217928665, −4.51351549924167581399763489475, −3.31296644192748182659080611566, −1.80172039958653432120378480743,
1.66458176566064431164403609055, 3.48875568387725635237668670730, 4.10921421989493223474440663100, 5.10874679156340899095832734268, 6.15008737905165369608162086921, 6.59679554498683474347203896074, 8.137404354408620170645329107161, 9.418827049596235342658372039413, 10.14733746166584988226188216206, 10.70716598974547888179567643079