| L(s) = 1 | − 2.08·5-s + 3.97·7-s − 5.31i·11-s + 4.83i·13-s + 6.52·17-s + 7.92·19-s − 1.37·23-s − 0.642·25-s + 3.16i·29-s + 5.86·31-s − 8.30·35-s − 8.85i·37-s − 7.58·41-s + 1.05i·43-s + 0.619i·47-s + ⋯ |
| L(s) = 1 | − 0.933·5-s + 1.50·7-s − 1.60i·11-s + 1.34i·13-s + 1.58·17-s + 1.81·19-s − 0.286·23-s − 0.128·25-s + 0.587i·29-s + 1.05·31-s − 1.40·35-s − 1.45i·37-s − 1.18·41-s + 0.160i·43-s + 0.0904i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278862229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278862229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + (5.22 + 11.8i)T \) |
| good | 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 5.31iT - 11T^{2} \) |
| 13 | \( 1 - 4.83iT - 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 3.16iT - 29T^{2} \) |
| 31 | \( 1 - 5.86T + 31T^{2} \) |
| 37 | \( 1 + 8.85iT - 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 - 1.05iT - 43T^{2} \) |
| 47 | \( 1 - 0.619iT - 47T^{2} \) |
| 53 | \( 1 + 0.405T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 6.00T + 71T^{2} \) |
| 73 | \( 1 + 3.66iT - 73T^{2} \) |
| 79 | \( 1 - 6.01iT - 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 - 3.50iT - 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.076413421956017732517022895280, −7.52869305720845157238930338507, −6.82734381922451819511141583766, −5.60858714034911911860133057803, −5.35040485328561832569936424673, −4.31352203920462559360516522575, −3.68571091829783479418159783266, −2.92904493039885969246154381595, −1.58202525391048744704032875761, −0.839601657877373538114445287714,
0.888693271527333735337359090014, 1.71886395149954701834995375221, 2.92916532065510133381550375553, 3.66369297665918949961788025214, 4.65739588909153196854043048113, 5.06160114328158596199038974152, 5.73746509584064580994335597062, 7.00145392000105532480276050668, 7.66994069487182027493739062303, 7.924767560108627203917497087731
