| L(s) = 1 | + 22.3·2-s − 27·3-s + 372.·4-s + 118.·5-s − 604.·6-s − 343·7-s + 5.46e3·8-s + 729·9-s + 2.66e3·10-s − 3.52e3·11-s − 1.00e4·12-s − 5.25e3·13-s − 7.67e3·14-s − 3.21e3·15-s + 7.46e4·16-s − 3.70e4·17-s + 1.63e4·18-s + 5.16e3·19-s + 4.43e4·20-s + 9.26e3·21-s − 7.87e4·22-s + 3.45e4·23-s − 1.47e5·24-s − 6.39e4·25-s − 1.17e5·26-s − 1.96e4·27-s − 1.27e5·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.90·4-s + 0.425·5-s − 1.14·6-s − 0.377·7-s + 3.77·8-s + 0.333·9-s + 0.841·10-s − 0.797·11-s − 1.67·12-s − 0.663·13-s − 0.747·14-s − 0.245·15-s + 4.55·16-s − 1.82·17-s + 0.659·18-s + 0.172·19-s + 1.23·20-s + 0.218·21-s − 1.57·22-s + 0.592·23-s − 2.18·24-s − 0.818·25-s − 1.31·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.197614882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.197614882\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 - 22.3T + 128T^{2} \) |
| 5 | \( 1 - 118.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.25e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.16e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.38e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.55e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.87e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.61e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.04e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.06e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−16.02811316544607560981621295880, −15.18851969542109728930663728516, −13.65206000825996643500787096170, −12.89709641092255898090487161303, −11.63743149679453847438288939971, −10.41394527918625825593600050991, −7.09346540608775711954501655338, −5.82581005654895952626718374074, −4.51729838368586493760445633859, −2.46334465876767302777906787572,
2.46334465876767302777906787572, 4.51729838368586493760445633859, 5.82581005654895952626718374074, 7.09346540608775711954501655338, 10.41394527918625825593600050991, 11.63743149679453847438288939971, 12.89709641092255898090487161303, 13.65206000825996643500787096170, 15.18851969542109728930663728516, 16.02811316544607560981621295880
