| L(s) = 1 | − 5-s − 5·11-s − 3·13-s + 3·17-s − 4·19-s − 7·23-s + 25-s + 8·29-s + 3·31-s − 3·41-s − 43-s − 7·49-s + 9·53-s + 5·55-s + 8·59-s + 3·65-s + 11·67-s + 8·71-s − 4·73-s + 8·79-s + 9·83-s − 3·85-s + 4·89-s + 4·95-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s − 0.832·13-s + 0.727·17-s − 0.917·19-s − 1.45·23-s + 1/5·25-s + 1.48·29-s + 0.538·31-s − 0.468·41-s − 0.152·43-s − 49-s + 1.23·53-s + 0.674·55-s + 1.04·59-s + 0.372·65-s + 1.34·67-s + 0.949·71-s − 0.468·73-s + 0.900·79-s + 0.987·83-s − 0.325·85-s + 0.423·89-s + 0.410·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.50048409900971, −14.76284075426217, −14.43965909098539, −13.73678731303046, −13.25381692240321, −12.64767420972625, −12.12302387505459, −11.86286289152017, −11.00468699960580, −10.48308959573782, −9.984780282496603, −9.725788156731119, −8.556353839017263, −8.301369983700698, −7.794091664109969, −7.241135651934334, −6.505779930120801, −5.956946416097138, −5.062196204582516, −4.870644435412997, −3.993372397131550, −3.342217128712832, −2.488209683809479, −2.135760207753469, −0.8252989054761261, 0,
0.8252989054761261, 2.135760207753469, 2.488209683809479, 3.342217128712832, 3.993372397131550, 4.870644435412997, 5.062196204582516, 5.956946416097138, 6.505779930120801, 7.241135651934334, 7.794091664109969, 8.301369983700698, 8.556353839017263, 9.725788156731119, 9.984780282496603, 10.48308959573782, 11.00468699960580, 11.86286289152017, 12.12302387505459, 12.64767420972625, 13.25381692240321, 13.73678731303046, 14.43965909098539, 14.76284075426217, 15.50048409900971
