| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 186.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s − 1.48e3·10-s − 1.33e3·11-s + 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 5.02e3·15-s + 4.09e3·16-s + 2.05e4·17-s − 5.83e3·18-s + 1.16e4·19-s + 1.19e4·20-s − 9.26e3·21-s + 1.06e4·22-s + 4.29e4·23-s − 1.38e4·24-s − 4.34e4·25-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 + 0.666·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.470·10-s − 0.302·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.384·15-s + 0.250·16-s + 1.01·17-s − 0.235·18-s + 0.390·19-s + 0.333·20-s − 0.218·21-s + 0.213·22-s + 0.736·23-s − 0.204·24-s − 0.556·25-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
| \(L(4)\) |
\(\approx\) |
\(2.358918951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.358918951\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 186.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.33e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.29e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.85e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.86e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.58e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.17e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.57e6T + 8.07e13T^{2} \) |
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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
−9.692213368501295558319806071163, −8.891579434502616864417878378873, −7.970466537218341566378529503841, −7.18994177767762460359144090577, −6.14121226468130288629094464152, −5.25197414379302353944380673993, −3.70436329117543321015842921995, −2.77333417597186984436927697719, −1.76116268641482602730359084769, −0.72896784250869987560480705609,
0.72896784250869987560480705609, 1.76116268641482602730359084769, 2.77333417597186984436927697719, 3.70436329117543321015842921995, 5.25197414379302353944380673993, 6.14121226468130288629094464152, 7.18994177767762460359144090577, 7.970466537218341566378529503841, 8.891579434502616864417878378873, 9.692213368501295558319806071163
