| L(s) = 1 | − 177.·3-s − 1.62e3·5-s + 1.83e3·7-s + 1.18e4·9-s + 3.17e4·11-s − 1.32e5·13-s + 2.87e5·15-s − 8.35e4·17-s + 1.60e3·19-s − 3.25e5·21-s − 2.32e4·23-s + 6.71e5·25-s + 1.39e6·27-s + 3.73e6·29-s − 8.91e6·31-s − 5.63e6·33-s − 2.97e6·35-s − 1.20e7·37-s + 2.34e7·39-s − 1.26e7·41-s − 2.86e7·43-s − 1.91e7·45-s + 7.17e6·47-s − 3.69e7·49-s + 1.48e7·51-s − 5.96e7·53-s − 5.14e7·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 1.15·5-s + 0.288·7-s + 0.599·9-s + 0.654·11-s − 1.28·13-s + 1.46·15-s − 0.242·17-s + 0.00282·19-s − 0.365·21-s − 0.0173·23-s + 0.343·25-s + 0.506·27-s + 0.980·29-s − 1.73·31-s − 0.827·33-s − 0.334·35-s − 1.05·37-s + 1.62·39-s − 0.700·41-s − 1.27·43-s − 0.695·45-s + 0.214·47-s − 0.916·49-s + 0.306·51-s − 1.03·53-s − 0.758·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1675242320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1675242320\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 8.35e4T \) |
| good | 3 | \( 1 + 177.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.62e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.32e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.60e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.32e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.20e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.26e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.86e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 7.17e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.96e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.85e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.27e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.48e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.58e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.89e8T + 7.60e17T^{2} \) |
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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
−10.65802253706128520533359989553, −9.472273545305115502850072889136, −8.255104922991919870508170211493, −7.26626191912924290109569638044, −6.46167702994012844521563218312, −5.18003978440908109615766464440, −4.51529846814718957394177994208, −3.29696398692107505753022360022, −1.60904669376382880009188016657, −0.19785623225894636513856731954,
0.19785623225894636513856731954, 1.60904669376382880009188016657, 3.29696398692107505753022360022, 4.51529846814718957394177994208, 5.18003978440908109615766464440, 6.46167702994012844521563218312, 7.26626191912924290109569638044, 8.255104922991919870508170211493, 9.472273545305115502850072889136, 10.65802253706128520533359989553
