| L(s) = 1 | + 38.8·2-s − 227.·3-s + 999.·4-s + 325.·5-s − 8.84e3·6-s + 1.89e4·8-s + 3.20e4·9-s + 1.26e4·10-s − 4.89e4·11-s − 2.27e5·12-s − 1.08e5·13-s − 7.40e4·15-s + 2.24e5·16-s − 2.28e5·17-s + 1.24e6·18-s − 5.68e4·19-s + 3.25e5·20-s − 1.90e6·22-s − 1.44e6·23-s − 4.30e6·24-s − 1.84e6·25-s − 4.22e6·26-s − 2.81e6·27-s − 2.66e5·29-s − 2.87e6·30-s + 1.61e5·31-s − 9.56e5·32-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 1.62·3-s + 1.95·4-s + 0.232·5-s − 2.78·6-s + 1.63·8-s + 1.62·9-s + 0.400·10-s − 1.00·11-s − 3.16·12-s − 1.05·13-s − 0.377·15-s + 0.858·16-s − 0.663·17-s + 2.79·18-s − 0.100·19-s + 0.454·20-s − 1.73·22-s − 1.07·23-s − 2.65·24-s − 0.945·25-s − 1.81·26-s − 1.01·27-s − 0.0699·29-s − 0.648·30-s + 0.0314·31-s − 0.161·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 38.8T + 512T^{2} \) |
| 3 | \( 1 + 227.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 325.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 4.89e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.08e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.28e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.68e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.44e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.66e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.61e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.08e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.73e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.91e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.00e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.56e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.47e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.22e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.56e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.62e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.10e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.75e8T + 7.60e17T^{2} \) |
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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
−12.86653739161800064732824739463, −12.13262633002850609458602090751, −11.20382147013038977754565572636, −10.15314350045654034017229356374, −7.30835195316444990636444078427, −6.07510977892417589526343411024, −5.30088719305749675611085789093, −4.29459726500826742135392331777, −2.28620735437153240085037496413, 0,
2.28620735437153240085037496413, 4.29459726500826742135392331777, 5.30088719305749675611085789093, 6.07510977892417589526343411024, 7.30835195316444990636444078427, 10.15314350045654034017229356374, 11.20382147013038977754565572636, 12.13262633002850609458602090751, 12.86653739161800064732824739463
