| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 498.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s − 3.99e3·10-s + 6.17e3·11-s + 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 1.34e4·15-s + 4.09e3·16-s + 6.82e3·17-s − 5.83e3·18-s − 1.94e4·19-s + 3.19e4·20-s − 9.26e3·21-s − 4.93e4·22-s + 4.13e3·23-s − 1.38e4·24-s + 1.70e5·25-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 + 1.78·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.26·10-s + 1.39·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1.03·15-s + 0.250·16-s + 0.336·17-s − 0.235·18-s − 0.648·19-s + 0.892·20-s − 0.218·21-s − 0.988·22-s + 0.0707·23-s − 0.204·24-s + 2.18·25-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
| \(L(4)\) |
\(\approx\) |
\(3.609116870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.609116870\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 498.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.17e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 6.82e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.94e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.13e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.29e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.23e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.52e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.31e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.47e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.99e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.29e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.86e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.38e7T + 8.07e13T^{2} \) |
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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
−9.629989324662486191650448977676, −8.992981744501017388324821087628, −8.198686979806622197586802237475, −6.64389668357971781193620737725, −6.47539399581337896796203843846, −5.26707762762996238223015351749, −3.77492868144901314025507558347, −2.60146306805393927797583587509, −1.74372157794805269828999231333, −0.948782190190546941993814076256,
0.948782190190546941993814076256, 1.74372157794805269828999231333, 2.60146306805393927797583587509, 3.77492868144901314025507558347, 5.26707762762996238223015351749, 6.47539399581337896796203843846, 6.64389668357971781193620737725, 8.198686979806622197586802237475, 8.992981744501017388324821087628, 9.629989324662486191650448977676
