| L(s) = 1 | + 13.7·2-s + 50.2·3-s + 59.9·4-s + 544.·5-s + 688.·6-s + 343·7-s − 933.·8-s + 336.·9-s + 7.46e3·10-s + 3.18e3·11-s + 3.00e3·12-s + 2.19e3·13-s + 4.70e3·14-s + 2.73e4·15-s − 2.04e4·16-s + 1.86e4·17-s + 4.60e3·18-s − 4.82e4·19-s + 3.26e4·20-s + 1.72e4·21-s + 4.36e4·22-s − 1.01e5·23-s − 4.68e4·24-s + 2.18e5·25-s + 3.01e4·26-s − 9.29e4·27-s + 2.05e4·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s + 1.07·3-s + 0.468·4-s + 1.94·5-s + 1.30·6-s + 0.377·7-s − 0.644·8-s + 0.153·9-s + 2.36·10-s + 0.721·11-s + 0.502·12-s + 0.277·13-s + 0.457·14-s + 2.09·15-s − 1.24·16-s + 0.918·17-s + 0.186·18-s − 1.61·19-s + 0.911·20-s + 0.405·21-s + 0.874·22-s − 1.73·23-s − 0.692·24-s + 2.79·25-s + 0.336·26-s − 0.908·27-s + 0.176·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.745233946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.745233946\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 13.7T + 128T^{2} \) |
| 3 | \( 1 - 50.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 544.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.18e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.01e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.63e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.69e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.11e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.16e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.78e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.67e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.03e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.19e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.66e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.87e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.39e6T + 8.07e13T^{2} \) |
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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
−13.15025155528877679010157463727, −12.07878168340253447018149409485, −10.35703635748690944880936873204, −9.300597993064132528130105027607, −8.446435104238127588247090558250, −6.39165253562428178466149435109, −5.65275245759701334397513400013, −4.18390529145104586327650728017, −2.75749677327559843177147731676, −1.77821097335086865608385862557,
1.77821097335086865608385862557, 2.75749677327559843177147731676, 4.18390529145104586327650728017, 5.65275245759701334397513400013, 6.39165253562428178466149435109, 8.446435104238127588247090558250, 9.300597993064132528130105027607, 10.35703635748690944880936873204, 12.07878168340253447018149409485, 13.15025155528877679010157463727
