| L(s) = 1 | + 21.5·3-s − 14.7·5-s + 49·7-s + 222.·9-s − 58.5·11-s + 1.17e3·13-s − 317.·15-s + 1.49e3·17-s − 498.·19-s + 1.05e3·21-s + 1.88e3·23-s − 2.90e3·25-s − 443.·27-s + 1.91e3·29-s − 794.·31-s − 1.26e3·33-s − 721.·35-s + 2.98e3·37-s + 2.54e4·39-s + 1.19e4·41-s − 9.82e3·43-s − 3.27e3·45-s + 1.96e4·47-s + 2.40e3·49-s + 3.22e4·51-s − 1.98e4·53-s + 862.·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s − 0.263·5-s + 0.377·7-s + 0.915·9-s − 0.145·11-s + 1.93·13-s − 0.364·15-s + 1.25·17-s − 0.317·19-s + 0.523·21-s + 0.744·23-s − 0.930·25-s − 0.117·27-s + 0.422·29-s − 0.148·31-s − 0.201·33-s − 0.0995·35-s + 0.358·37-s + 2.67·39-s + 1.10·41-s − 0.809·43-s − 0.241·45-s + 1.29·47-s + 0.142·49-s + 1.73·51-s − 0.971·53-s + 0.0384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.166156764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.166156764\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 21.5T + 243T^{2} \) |
| 5 | \( 1 + 14.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 58.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 498.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 794.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.35e4T + 8.58e9T^{2} \) |
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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
−13.00265354754268158607019121227, −11.61937122265126939252123601461, −10.49773667111504950591165934186, −9.175644737350158140734299292197, −8.352709442719616720713951114048, −7.56753035858798035479234531490, −5.89544904508370381566828272981, −4.05904731979819345079837105692, −3.03014646403220171801072221759, −1.38719767701399037352558598127,
1.38719767701399037352558598127, 3.03014646403220171801072221759, 4.05904731979819345079837105692, 5.89544904508370381566828272981, 7.56753035858798035479234531490, 8.352709442719616720713951114048, 9.175644737350158140734299292197, 10.49773667111504950591165934186, 11.61937122265126939252123601461, 13.00265354754268158607019121227
