| L(s) = 1 | + 2.76·2-s + 3-s + 5.62·4-s + 2.76·6-s − 1.86·7-s + 10.0·8-s + 9-s + 11-s + 5.62·12-s − 4.62·13-s − 5.14·14-s + 16.4·16-s − 2.49·17-s + 2.76·18-s − 5.38·19-s − 1.86·21-s + 2.76·22-s − 7.14·23-s + 10.0·24-s − 12.7·26-s + 27-s − 10.4·28-s − 3.52·29-s + 8.62·31-s + 25.2·32-s + 33-s − 6.87·34-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.577·3-s + 2.81·4-s + 1.12·6-s − 0.704·7-s + 3.54·8-s + 0.333·9-s + 0.301·11-s + 1.62·12-s − 1.28·13-s − 1.37·14-s + 4.10·16-s − 0.604·17-s + 0.650·18-s − 1.23·19-s − 0.406·21-s + 0.588·22-s − 1.49·23-s + 2.04·24-s − 2.50·26-s + 0.192·27-s − 1.98·28-s − 0.654·29-s + 1.54·31-s + 4.46·32-s + 0.174·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.564785285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.564785285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 + 3.52T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 0.761T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 - 0.373T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 6.77T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 9.04T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−10.36014159722245214479663626176, −9.667369585063204447880372594028, −8.223247391396777714175356612397, −7.28489429468389571638328499240, −6.50659426085273659986767504246, −5.80716878558180185936144345946, −4.49763813187114139855412570437, −4.05657716670590664058501119166, −2.80874805352359110110031403542, −2.14848788018557771862822690356,
2.14848788018557771862822690356, 2.80874805352359110110031403542, 4.05657716670590664058501119166, 4.49763813187114139855412570437, 5.80716878558180185936144345946, 6.50659426085273659986767504246, 7.28489429468389571638328499240, 8.223247391396777714175356612397, 9.667369585063204447880372594028, 10.36014159722245214479663626176
