| L(s) = 1 | − 5-s − 2·7-s + 11-s + 7·17-s − 7·19-s − 4·23-s + 25-s − 6·29-s − 8·31-s + 2·35-s − 7·37-s + 3·41-s − 9·43-s + 7·47-s − 3·49-s + 6·53-s − 55-s + 14·59-s + 2·61-s + 5·67-s + 2·71-s − 8·73-s − 2·77-s + 4·79-s − 7·85-s + 16·89-s + 7·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.301·11-s + 1.69·17-s − 1.60·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 1.15·37-s + 0.468·41-s − 1.37·43-s + 1.02·47-s − 3/7·49-s + 0.824·53-s − 0.134·55-s + 1.82·59-s + 0.256·61-s + 0.610·67-s + 0.237·71-s − 0.936·73-s − 0.227·77-s + 0.450·79-s − 0.759·85-s + 1.69·89-s + 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.73050546833759, −12.48062948378516, −11.83155955251363, −11.63578972860918, −10.96782713924316, −10.42998777741061, −10.16927878529951, −9.696103574547688, −9.085324940965014, −8.765344118136974, −8.204537966687317, −7.743665601116727, −7.257558286967771, −6.819051522537026, −6.309657025971293, −5.772326548807966, −5.388864525413237, −4.832458324555166, −3.990961000572185, −3.646037662047815, −3.520651572836266, −2.587671455118863, −2.045451907361592, −1.477507698752031, −0.6001552328774249, 0,
0.6001552328774249, 1.477507698752031, 2.045451907361592, 2.587671455118863, 3.520651572836266, 3.646037662047815, 3.990961000572185, 4.832458324555166, 5.388864525413237, 5.772326548807966, 6.309657025971293, 6.819051522537026, 7.257558286967771, 7.743665601116727, 8.204537966687317, 8.765344118136974, 9.085324940965014, 9.696103574547688, 10.16927878529951, 10.42998777741061, 10.96782713924316, 11.63578972860918, 11.83155955251363, 12.48062948378516, 12.73050546833759
