| L(s) = 1 | + 21.5·2-s + 27·3-s + 336.·4-s − 418.·5-s + 581.·6-s + 343·7-s + 4.48e3·8-s + 729·9-s − 9.01e3·10-s − 3.90e3·11-s + 9.07e3·12-s + 7.55e3·13-s + 7.38e3·14-s − 1.13e4·15-s + 5.35e4·16-s + 1.31e4·17-s + 1.57e4·18-s + 9.20e3·19-s − 1.40e5·20-s + 9.26e3·21-s − 8.40e4·22-s − 1.21e4·23-s + 1.20e5·24-s + 9.70e4·25-s + 1.62e5·26-s + 1.96e4·27-s + 1.15e5·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.62·4-s − 1.49·5-s + 1.09·6-s + 0.377·7-s + 3.09·8-s + 0.333·9-s − 2.85·10-s − 0.884·11-s + 1.51·12-s + 0.953·13-s + 0.719·14-s − 0.864·15-s + 3.26·16-s + 0.651·17-s + 0.634·18-s + 0.307·19-s − 3.93·20-s + 0.218·21-s − 1.68·22-s − 0.208·23-s + 1.78·24-s + 1.24·25-s + 1.81·26-s + 0.192·27-s + 0.992·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.313961984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.313961984\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 21.5T + 128T^{2} \) |
| 5 | \( 1 + 418.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.55e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.20e3T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.06e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.24e3T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.19e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.02e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.06e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.75e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.92e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.56e6T + 8.07e13T^{2} \) |
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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
−10.32984423035410333049156264221, −8.454378370207508310339673008833, −7.77360709128908354072635137025, −7.06951560111443028007591341636, −5.86726191266298375928547180452, −4.84123359412171531016051018592, −4.04704797244716465895499436338, −3.35328034513637512574396998168, −2.49197603386544172144946024152, −1.02956700030289055012797134930,
1.02956700030289055012797134930, 2.49197603386544172144946024152, 3.35328034513637512574396998168, 4.04704797244716465895499436338, 4.84123359412171531016051018592, 5.86726191266298375928547180452, 7.06951560111443028007591341636, 7.77360709128908354072635137025, 8.454378370207508310339673008833, 10.32984423035410333049156264221
