| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 262.·5-s − 216·6-s + 343·7-s + 512·8-s + 729·9-s − 2.09e3·10-s + 8.08e3·11-s − 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 7.08e3·15-s + 4.09e3·16-s + 3.63e4·17-s + 5.83e3·18-s + 7.94e3·19-s − 1.67e4·20-s − 9.26e3·21-s + 6.47e4·22-s + 8.04e4·23-s − 1.38e4·24-s − 9.35e3·25-s − 1.75e4·26-s − 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.938·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.663·10-s + 1.83·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.541·15-s + 0.250·16-s + 1.79·17-s + 0.235·18-s + 0.265·19-s − 0.469·20-s − 0.218·21-s + 1.29·22-s + 1.37·23-s − 0.204·24-s − 0.119·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.224642162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224642162\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 + 262.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 8.08e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 7.94e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.04e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.89e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.15e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.10e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.31e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.45e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.58e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.75e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.74e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.14e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.12e7T + 8.07e13T^{2} \) |
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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
−9.732796760718715979987023283540, −8.777690392652702000524845628068, −7.45736192931687795355533536589, −7.08385219765376205430754759343, −5.82387522649818531177747831026, −5.07168582464968601027612626028, −3.90958205808362665305146842822, −3.45493344644848667042509871548, −1.67110483367858199849726440604, −0.76601322036625800602058739366,
0.76601322036625800602058739366, 1.67110483367858199849726440604, 3.45493344644848667042509871548, 3.90958205808362665305146842822, 5.07168582464968601027612626028, 5.82387522649818531177747831026, 7.08385219765376205430754759343, 7.45736192931687795355533536589, 8.777690392652702000524845628068, 9.732796760718715979987023283540
