| L(s) = 1 | + 2.68·2-s + 9·3-s − 24.7·4-s − 83.5·5-s + 24.2·6-s − 48.3·7-s − 152.·8-s + 81·9-s − 224.·10-s + 19.9·11-s − 222.·12-s + 611.·13-s − 130.·14-s − 751.·15-s + 381.·16-s + 1.52e3·17-s + 217.·18-s − 1.27e3·19-s + 2.06e3·20-s − 435.·21-s + 53.7·22-s + 3.30e3·23-s − 1.37e3·24-s + 3.84e3·25-s + 1.64e3·26-s + 729·27-s + 1.19e3·28-s + ⋯ |
| L(s) = 1 | + 0.475·2-s + 0.577·3-s − 0.773·4-s − 1.49·5-s + 0.274·6-s − 0.372·7-s − 0.843·8-s + 0.333·9-s − 0.710·10-s + 0.0498·11-s − 0.446·12-s + 1.00·13-s − 0.177·14-s − 0.862·15-s + 0.373·16-s + 1.28·17-s + 0.158·18-s − 0.808·19-s + 1.15·20-s − 0.215·21-s + 0.0236·22-s + 1.30·23-s − 0.486·24-s + 1.23·25-s + 0.477·26-s + 0.192·27-s + 0.288·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.660761036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660761036\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 2.68T + 32T^{2} \) |
| 5 | \( 1 + 83.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 48.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 19.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 611.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.52e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.04e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 134.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.27e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.73e4T + 8.58e9T^{2} \) |
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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
−12.06227576188018370867229600531, −10.93903361686578287070498589384, −9.632720324576279990524393285943, −8.542212768814982843045226158648, −7.955181282823108552249359886338, −6.56671850162300701319034474384, −4.99809879787258111609107788425, −3.82604256304045866174365312928, −3.25187533875933838619423325117, −0.75301853343308106950638934007,
0.75301853343308106950638934007, 3.25187533875933838619423325117, 3.82604256304045866174365312928, 4.99809879787258111609107788425, 6.56671850162300701319034474384, 7.955181282823108552249359886338, 8.542212768814982843045226158648, 9.632720324576279990524393285943, 10.93903361686578287070498589384, 12.06227576188018370867229600531
