| L(s) = 1 | − 9·3-s + 71.6·5-s − 49·7-s + 81·9-s − 567.·11-s + 831.·13-s − 644.·15-s + 888.·17-s + 2.91e3·19-s + 441·21-s + 3.10e3·23-s + 2.00e3·25-s − 729·27-s + 8.27e3·29-s − 7.02e3·31-s + 5.10e3·33-s − 3.50e3·35-s − 1.01e4·37-s − 7.48e3·39-s + 3.09e3·41-s + 1.50e4·43-s + 5.80e3·45-s + 1.98e4·47-s + 2.40e3·49-s − 7.99e3·51-s − 9.20e3·53-s − 4.06e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.28·5-s − 0.377·7-s + 0.333·9-s − 1.41·11-s + 1.36·13-s − 0.739·15-s + 0.745·17-s + 1.85·19-s + 0.218·21-s + 1.22·23-s + 0.641·25-s − 0.192·27-s + 1.82·29-s − 1.31·31-s + 0.816·33-s − 0.484·35-s − 1.21·37-s − 0.788·39-s + 0.287·41-s + 1.23·43-s + 0.427·45-s + 1.31·47-s + 0.142·49-s − 0.430·51-s − 0.450·53-s − 1.81·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.836901842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836901842\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 - 71.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 567.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 831.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 888.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.98e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.20e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.41e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.66e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.80e4T + 8.58e9T^{2} \) |
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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
−13.37449393757759025822871781524, −12.36927526074276033777729526304, −10.88637087035719379233582877228, −10.14131161289535531956222982294, −9.020374215300732690055887067713, −7.38488820157283642554665432599, −5.94729515074259491783820941558, −5.24757818509928590157425842175, −3.02176397617772128275971656079, −1.14117276929874442954233315634,
1.14117276929874442954233315634, 3.02176397617772128275971656079, 5.24757818509928590157425842175, 5.94729515074259491783820941558, 7.38488820157283642554665432599, 9.020374215300732690055887067713, 10.14131161289535531956222982294, 10.88637087035719379233582877228, 12.36927526074276033777729526304, 13.37449393757759025822871781524
