| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 11-s + 2·13-s + 4·14-s + 16-s + 17-s + 2·19-s + 20-s − 22-s − 6·23-s + 25-s − 2·26-s − 4·28-s + 6·29-s − 4·31-s − 32-s − 34-s − 4·35-s − 10·37-s − 2·38-s − 40-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.458·19-s + 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.676·35-s − 1.64·37-s − 0.324·38-s − 0.158·40-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.16944179553488, −15.67536027211518, −15.47599014148826, −14.42519750353164, −13.82728852954795, −13.60513249405756, −12.61296302851540, −12.36422533550998, −11.79041122750328, −10.90882787069084, −10.42321069158495, −9.868737500513542, −9.486831157405747, −8.870964307650077, −8.327052498090627, −7.551834081153248, −6.783835025778869, −6.488158942514709, −5.804492736850668, −5.233959546285740, −4.006391451977806, −3.492422036682214, −2.769932241361962, −1.942478839429671, −1.006454268236412, 0,
1.006454268236412, 1.942478839429671, 2.769932241361962, 3.492422036682214, 4.006391451977806, 5.233959546285740, 5.804492736850668, 6.488158942514709, 6.783835025778869, 7.551834081153248, 8.327052498090627, 8.870964307650077, 9.486831157405747, 9.868737500513542, 10.42321069158495, 10.90882787069084, 11.79041122750328, 12.36422533550998, 12.61296302851540, 13.60513249405756, 13.82728852954795, 14.42519750353164, 15.47599014148826, 15.67536027211518, 16.16944179553488
