| L(s) = 1 | + 20.1·3-s − 25·5-s + 162.·9-s − 427.·11-s + 646.·13-s − 503.·15-s + 1.02e3·17-s − 2.25e3·19-s + 2.59e3·23-s + 625·25-s − 1.62e3·27-s + 869.·29-s − 4.40e3·31-s − 8.59e3·33-s + 2.55e3·37-s + 1.30e4·39-s − 6.22e3·41-s + 7.75e3·43-s − 4.05e3·45-s − 1.88e3·47-s + 2.07e4·51-s − 1.17e4·53-s + 1.06e4·55-s − 4.53e4·57-s − 3.33e4·59-s + 1.21e3·61-s − 1.61e4·65-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 0.447·5-s + 0.666·9-s − 1.06·11-s + 1.06·13-s − 0.577·15-s + 0.863·17-s − 1.43·19-s + 1.02·23-s + 0.200·25-s − 0.430·27-s + 0.192·29-s − 0.823·31-s − 1.37·33-s + 0.306·37-s + 1.37·39-s − 0.578·41-s + 0.639·43-s − 0.298·45-s − 0.124·47-s + 1.11·51-s − 0.573·53-s + 0.475·55-s − 1.84·57-s − 1.24·59-s + 0.0417·61-s − 0.474·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 20.1T + 243T^{2} \) |
| 11 | \( 1 + 427.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 646.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 869.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.22e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.21e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.17e4T + 8.58e9T^{2} \) |
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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
−8.725747667662136187707887664478, −8.086004655283926541678364227659, −7.52045329416865492162302122355, −6.38956941735003558885943973471, −5.30415547691790737388658472838, −4.15936849695453949925996211234, −3.31546586434693942001550798589, −2.58120375359014087902261453227, −1.42669487291980306787490487317, 0,
1.42669487291980306787490487317, 2.58120375359014087902261453227, 3.31546586434693942001550798589, 4.15936849695453949925996211234, 5.30415547691790737388658472838, 6.38956941735003558885943973471, 7.52045329416865492162302122355, 8.086004655283926541678364227659, 8.725747667662136187707887664478
