| L(s) = 1 | + 18.9·2-s + 232·4-s − 303.·5-s + 260·7-s + 1.97e3·8-s − 5.76e3·10-s − 6.07e3·11-s + 6.89e3·13-s + 4.93e3·14-s + 7.74e3·16-s + 2.36e4·17-s + 3.31e4·19-s − 7.04e4·20-s − 1.15e5·22-s + 3.15e4·23-s + 1.40e4·25-s + 1.30e5·26-s + 6.03e4·28-s − 1.38e5·29-s + 1.50e3·31-s − 1.05e5·32-s + 4.49e5·34-s − 7.89e4·35-s − 3.80e5·37-s + 6.29e5·38-s − 5.99e5·40-s + 8.80e4·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.81·4-s − 1.08·5-s + 0.286·7-s + 1.36·8-s − 1.82·10-s − 1.37·11-s + 0.869·13-s + 0.480·14-s + 0.472·16-s + 1.16·17-s + 1.10·19-s − 1.96·20-s − 2.30·22-s + 0.541·23-s + 0.179·25-s + 1.45·26-s + 0.519·28-s − 1.05·29-s + 0.00909·31-s − 0.569·32-s + 1.96·34-s − 0.311·35-s − 1.23·37-s + 1.86·38-s − 1.47·40-s + 0.199·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.674512201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.674512201\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 18.9T + 128T^{2} \) |
| 5 | \( 1 + 303.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 260T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.89e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.80e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.64e3T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.65e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.70e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.88e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.00e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.29e7T + 8.07e13T^{2} \) |
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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
−20.38969099516560214293303929997, −18.62599180950191572881612873522, −16.09721304205976252787375380795, −15.22933754465762612426156489372, −13.72863634433878072038780045570, −12.34634256857787230273205608094, −11.08169161750989189708347039487, −7.64730526586133418050975846402, −5.31877812668296384837894467469, −3.45528160667815042061928596501,
3.45528160667815042061928596501, 5.31877812668296384837894467469, 7.64730526586133418050975846402, 11.08169161750989189708347039487, 12.34634256857787230273205608094, 13.72863634433878072038780045570, 15.22933754465762612426156489372, 16.09721304205976252787375380795, 18.62599180950191572881612873522, 20.38969099516560214293303929997
