| L(s) = 1 | + 0.774i·5-s + 1.59i·7-s − 2.37·11-s − 3.72·13-s + 7.33i·17-s + 3.66i·19-s + 7.80·23-s + 4.39·25-s − 1.33i·29-s + 6.39i·31-s − 1.23·35-s − 37-s − 7.60i·41-s − 3.09i·43-s − 5.12·47-s + ⋯ |
| L(s) = 1 | + 0.346i·5-s + 0.604i·7-s − 0.716·11-s − 1.03·13-s + 1.77i·17-s + 0.839i·19-s + 1.62·23-s + 0.879·25-s − 0.248i·29-s + 1.14i·31-s − 0.209·35-s − 0.164·37-s − 1.18i·41-s − 0.471i·43-s − 0.746·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8017260398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8017260398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 0.774iT - 5T^{2} \) |
| 7 | \( 1 - 1.59iT - 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 7.33iT - 17T^{2} \) |
| 19 | \( 1 - 3.66iT - 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 + 1.33iT - 29T^{2} \) |
| 31 | \( 1 - 6.39iT - 31T^{2} \) |
| 41 | \( 1 + 7.60iT - 41T^{2} \) |
| 43 | \( 1 + 3.09iT - 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + 8.58iT - 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 + 2.99iT - 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 + 8.36T + 83T^{2} \) |
| 89 | \( 1 - 7.82iT - 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.501154044216565048714095278648, −7.88297859227547117251707753911, −7.03135705889522907476312438891, −6.49574459134595827754347743302, −5.46065571793367112776267464097, −5.13851188525119806489807222512, −4.04746417326955267209888550515, −3.15325284312566825032671265469, −2.43290067389607021194046036626, −1.45855941101337150465197865445,
0.22125119298995941193129651980, 1.17026437073991616867311596017, 2.73821696513331252360306196738, 2.91647670710112174649572931835, 4.45069636418885142391625034892, 4.83106026630590193534091590456, 5.43367751902194287335375692813, 6.60175000249969075613108303438, 7.29853120661260732507228850032, 7.56922653520728073740751151708
