| L(s) = 1 | + 40.7·2-s + 136.·3-s + 1.14e3·4-s − 1.67e3·5-s + 5.57e3·6-s + 6.27e3·7-s + 2.58e4·8-s − 934.·9-s − 6.81e4·10-s + 5.93e4·11-s + 1.57e5·12-s + 1.29e5·13-s + 2.55e5·14-s − 2.29e5·15-s + 4.66e5·16-s + 3.44e5·17-s − 3.80e4·18-s − 8.05e5·19-s − 1.91e6·20-s + 8.59e5·21-s + 2.41e6·22-s − 1.28e6·23-s + 3.54e6·24-s + 8.44e5·25-s + 5.26e6·26-s − 2.82e6·27-s + 7.19e6·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.975·3-s + 2.24·4-s − 1.19·5-s + 1.75·6-s + 0.987·7-s + 2.23·8-s − 0.0475·9-s − 2.15·10-s + 1.22·11-s + 2.18·12-s + 1.25·13-s + 1.77·14-s − 1.16·15-s + 1.77·16-s + 1.00·17-s − 0.0855·18-s − 1.41·19-s − 2.68·20-s + 0.963·21-s + 2.20·22-s − 0.957·23-s + 2.17·24-s + 0.432·25-s + 2.25·26-s − 1.02·27-s + 2.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.116873978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.116873978\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 40.7T + 512T^{2} \) |
| 3 | \( 1 - 136.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.67e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.29e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.28e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.69e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.77e6T + 3.27e14T^{2} \) |
| 47 | \( 1 - 3.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.20e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.61e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.96e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.14e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.61e8T + 7.60e17T^{2} \) |
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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
−14.33770153075273306744678885571, −13.00707884580634582305173433017, −11.74720389556662786812203858131, −11.15572068862501784352186181626, −8.600968749659464982159061787083, −7.53540780951769973031193026167, −5.89423687819400493355795369894, −4.06949092422323627082442959183, −3.64208594448593197260798577085, −1.85289306595340552836891836546,
1.85289306595340552836891836546, 3.64208594448593197260798577085, 4.06949092422323627082442959183, 5.89423687819400493355795369894, 7.53540780951769973031193026167, 8.600968749659464982159061787083, 11.15572068862501784352186181626, 11.74720389556662786812203858131, 13.00707884580634582305173433017, 14.33770153075273306744678885571
