| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 3·9-s − 2·13-s − 2·14-s + 16-s + 6·17-s + 3·18-s − 6·19-s − 4·23-s + 2·26-s + 2·28-s − 4·31-s − 32-s − 6·34-s − 3·36-s + 37-s + 6·38-s − 10·41-s + 4·43-s + 4·46-s + 2·47-s − 3·49-s − 2·52-s − 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 9-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s − 0.834·23-s + 0.392·26-s + 0.377·28-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s + 0.164·37-s + 0.973·38-s − 1.56·41-s + 0.609·43-s + 0.589·46-s + 0.291·47-s − 3/7·49-s − 0.277·52-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.776464432479204247972797831801, −8.062677850001302204457856503347, −7.61843478274367654964811575512, −6.48983703143827331517471330308, −5.71184317840727757217829967484, −4.89537956640686110087732549653, −3.67934155458315654719216021516, −2.59022325384696380905670929706, −1.59496787970180487849418213093, 0,
1.59496787970180487849418213093, 2.59022325384696380905670929706, 3.67934155458315654719216021516, 4.89537956640686110087732549653, 5.71184317840727757217829967484, 6.48983703143827331517471330308, 7.61843478274367654964811575512, 8.062677850001302204457856503347, 8.776464432479204247972797831801
