| L(s) = 1 | + 16.6·2-s + 150.·4-s − 125·5-s + 1.29e3·7-s + 369.·8-s − 2.08e3·10-s − 3.09e3·11-s − 6.58e3·13-s + 2.15e4·14-s − 1.30e4·16-s + 2.96e3·17-s + 1.63e4·19-s − 1.87e4·20-s − 5.15e4·22-s − 5.36e4·23-s + 1.56e4·25-s − 1.09e5·26-s + 1.93e5·28-s + 6.50e3·29-s + 2.78e4·31-s − 2.65e5·32-s + 4.95e4·34-s − 1.61e5·35-s + 1.87e5·37-s + 2.71e5·38-s − 4.61e4·40-s − 7.45e5·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s − 0.447·5-s + 1.42·7-s + 0.254·8-s − 0.659·10-s − 0.700·11-s − 0.831·13-s + 2.09·14-s − 0.797·16-s + 0.146·17-s + 0.545·19-s − 0.524·20-s − 1.03·22-s − 0.919·23-s + 0.199·25-s − 1.22·26-s + 1.66·28-s + 0.0495·29-s + 0.168·31-s − 1.43·32-s + 0.216·34-s − 0.636·35-s + 0.609·37-s + 0.803·38-s − 0.113·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 16.6T + 128T^{2} \) |
| 7 | \( 1 - 1.29e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.58e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.96e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.63e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.36e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.50e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.78e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.87e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.18e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.91e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.73e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.22e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 6.41e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.29e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.49e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.04e7T + 8.07e13T^{2} \) |
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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
−9.817793456846437924222511633427, −8.362049081124922310640385680695, −7.69932375446926510392254639714, −6.60756235740169826753904484348, −5.25020308969558435379036039789, −4.94030366653406018033693259161, −3.90623742707429563932651148693, −2.78828568071013044237769438307, −1.70968267315000397209760514108, 0,
1.70968267315000397209760514108, 2.78828568071013044237769438307, 3.90623742707429563932651148693, 4.94030366653406018033693259161, 5.25020308969558435379036039789, 6.60756235740169826753904484348, 7.69932375446926510392254639714, 8.362049081124922310640385680695, 9.817793456846437924222511633427
