| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 4·13-s − 14-s + 16-s + 6·17-s + 19-s − 20-s + 25-s + 4·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s − 35-s + 2·37-s − 38-s + 40-s − 6·41-s − 4·43-s − 6·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s − 0.162·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.68636078926163, −16.28370416511863, −15.55753475506016, −14.99090482292782, −14.51885444600664, −14.03155665505039, −13.18583398399555, −12.35721409844886, −12.08146342674333, −11.54267512440099, −10.81627967596889, −10.20609828644529, −9.746850868419354, −9.121833743039352, −8.319551263690301, −7.859583624790396, −7.381008149223477, −6.713764508571020, −5.915326259436854, −5.093393515487466, −4.613016946081498, −3.477998465948550, −2.963334336772322, −1.947242698174730, −1.098111227582373, 0,
1.098111227582373, 1.947242698174730, 2.963334336772322, 3.477998465948550, 4.613016946081498, 5.093393515487466, 5.915326259436854, 6.713764508571020, 7.381008149223477, 7.859583624790396, 8.319551263690301, 9.121833743039352, 9.746850868419354, 10.20609828644529, 10.81627967596889, 11.54267512440099, 12.08146342674333, 12.35721409844886, 13.18583398399555, 14.03155665505039, 14.51885444600664, 14.99090482292782, 15.55753475506016, 16.28370416511863, 16.68636078926163
