| L(s) = 1 | − 33.1·2-s − 81·3-s + 584.·4-s + 625·5-s + 2.68e3·6-s + 17.0·7-s − 2.40e3·8-s + 6.56e3·9-s − 2.06e4·10-s − 3.84e4·11-s − 4.73e4·12-s − 7.99e4·13-s − 564.·14-s − 5.06e4·15-s − 2.19e5·16-s − 1.84e5·17-s − 2.17e5·18-s − 1.30e5·19-s + 3.65e5·20-s − 1.37e3·21-s + 1.27e6·22-s − 2.31e6·23-s + 1.94e5·24-s + 3.90e5·25-s + 2.64e6·26-s − 5.31e5·27-s + 9.95e3·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.577·3-s + 1.14·4-s + 0.447·5-s + 0.844·6-s + 0.00268·7-s − 0.207·8-s + 0.333·9-s − 0.654·10-s − 0.790·11-s − 0.659·12-s − 0.776·13-s − 0.00392·14-s − 0.258·15-s − 0.838·16-s − 0.536·17-s − 0.487·18-s − 0.229·19-s + 0.510·20-s − 0.00154·21-s + 1.15·22-s − 1.72·23-s + 0.119·24-s + 0.200·25-s + 1.13·26-s − 0.192·27-s + 0.00306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2336922185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2336922185\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 33.1T + 512T^{2} \) |
| 7 | \( 1 - 17.0T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.84e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.99e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.84e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 2.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.71e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.21e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.02e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.88e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.39e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.46e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.64e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.59e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.94e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.16e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.51e8T + 7.60e17T^{2} \) |
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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
−10.09630933122325447432559060297, −9.505418474036093931870279565129, −8.377623380954622209377067577865, −7.57039473260007830518591711921, −6.61375560135609847054800602560, −5.51538809323591965944431657831, −4.35255281293163704531633946412, −2.47903219853053816247996425429, −1.61900788141202893665447400164, −0.27497197698795758052291232934,
0.27497197698795758052291232934, 1.61900788141202893665447400164, 2.47903219853053816247996425429, 4.35255281293163704531633946412, 5.51538809323591965944431657831, 6.61375560135609847054800602560, 7.57039473260007830518591711921, 8.377623380954622209377067577865, 9.505418474036093931870279565129, 10.09630933122325447432559060297
