| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 21·5-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s − 84·10-s − 654·11-s + 144·12-s + 169·13-s + 196·14-s − 189·15-s + 256·16-s + 756·17-s + 324·18-s − 2.90e3·19-s − 336·20-s + 441·21-s − 2.61e3·22-s + 3.03e3·23-s + 576·24-s − 2.68e3·25-s + 676·26-s + 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.375·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.265·10-s − 1.62·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.216·15-s + 1/4·16-s + 0.634·17-s + 0.235·18-s − 1.84·19-s − 0.187·20-s + 0.218·21-s − 1.15·22-s + 1.19·23-s + 0.204·24-s − 0.858·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 21 T + p^{5} T^{2} \) |
| 11 | \( 1 + 654 T + p^{5} T^{2} \) |
| 17 | \( 1 - 756 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2905 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3039 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3957 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3241 T + p^{5} T^{2} \) |
| 37 | \( 1 + 628 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3678 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8797 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14163 T + p^{5} T^{2} \) |
| 53 | \( 1 + 16497 T + p^{5} T^{2} \) |
| 59 | \( 1 + 24528 T + p^{5} T^{2} \) |
| 61 | \( 1 - 3686 T + p^{5} T^{2} \) |
| 67 | \( 1 - 63818 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14040 T + p^{5} T^{2} \) |
| 73 | \( 1 + 4579 T + p^{5} T^{2} \) |
| 79 | \( 1 + 20713 T + p^{5} T^{2} \) |
| 83 | \( 1 - 59421 T + p^{5} T^{2} \) |
| 89 | \( 1 + 112995 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78511 T + p^{5} T^{2} \) |
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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
−9.634589211314560288750981874179, −8.367195928802873099197824271778, −7.87467732537240304099694528842, −6.90982044636884497850152803434, −5.67732094513245775535797074392, −4.80069516032433837116972911289, −3.78480267581404163569355581877, −2.77293551516311811652408095102, −1.76059048240268090775901869359, 0,
1.76059048240268090775901869359, 2.77293551516311811652408095102, 3.78480267581404163569355581877, 4.80069516032433837116972911289, 5.67732094513245775535797074392, 6.90982044636884497850152803434, 7.87467732537240304099694528842, 8.367195928802873099197824271778, 9.634589211314560288750981874179
