| L(s) = 1 | + 5-s + 4·7-s + 11-s + 13-s − 4·17-s + 6·19-s − 2·23-s + 25-s − 6·29-s − 2·31-s + 4·35-s − 8·37-s − 2·41-s − 8·43-s + 9·49-s − 12·53-s + 55-s − 12·59-s − 14·61-s + 65-s + 16·67-s + 12·71-s + 4·73-s + 4·77-s + 16·79-s + 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.277·13-s − 0.970·17-s + 1.37·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.676·35-s − 1.31·37-s − 0.312·41-s − 1.21·43-s + 9/7·49-s − 1.64·53-s + 0.134·55-s − 1.56·59-s − 1.79·61-s + 0.124·65-s + 1.95·67-s + 1.42·71-s + 0.468·73-s + 0.455·77-s + 1.80·79-s + 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.77909119690900, −14.05439163377299, −13.77850676723721, −13.52345261742888, −12.54664135931830, −12.23419809426371, −11.54935079504562, −11.06291514256824, −10.89246522697299, −10.10004388048492, −9.393318556492147, −9.138371984829740, −8.446190273069021, −7.827417902817942, −7.592413834721200, −6.645090467405651, −6.372484662373269, −5.315854128386520, −5.204686210577887, −4.583162628784883, −3.758339912639363, −3.266827199339204, −2.214827301863700, −1.735076995017091, −1.221210341344202, 0,
1.221210341344202, 1.735076995017091, 2.214827301863700, 3.266827199339204, 3.758339912639363, 4.583162628784883, 5.204686210577887, 5.315854128386520, 6.372484662373269, 6.645090467405651, 7.592413834721200, 7.827417902817942, 8.446190273069021, 9.138371984829740, 9.393318556492147, 10.10004388048492, 10.89246522697299, 11.06291514256824, 11.54935079504562, 12.23419809426371, 12.54664135931830, 13.52345261742888, 13.77850676723721, 14.05439163377299, 14.77909119690900
