| L(s) = 1 | − 3·5-s + 7-s + 4·11-s − 3·13-s + 4·17-s + 23-s + 4·25-s − 3·29-s + 6·31-s − 3·35-s − 9·37-s − 9·41-s + 3·43-s − 7·47-s + 49-s + 4·53-s − 12·55-s + 6·59-s + 10·61-s + 9·65-s − 4·67-s − 6·71-s − 8·73-s + 4·77-s − 8·79-s + 4·83-s − 12·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 1.20·11-s − 0.832·13-s + 0.970·17-s + 0.208·23-s + 4/5·25-s − 0.557·29-s + 1.07·31-s − 0.507·35-s − 1.47·37-s − 1.40·41-s + 0.457·43-s − 1.02·47-s + 1/7·49-s + 0.549·53-s − 1.61·55-s + 0.781·59-s + 1.28·61-s + 1.11·65-s − 0.488·67-s − 0.712·71-s − 0.936·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.74705533645298, −15.05417739668952, −14.74356554035622, −14.32650848925906, −13.65301710422316, −12.95459830776578, −12.20114361064168, −11.87086873304509, −11.65334453143358, −10.97682072108973, −10.09781624537977, −9.891258623986637, −8.817100133806418, −8.627637395954805, −7.833961561680686, −7.377333340246940, −6.882959217382957, −6.191339648462303, −5.246798714692609, −4.814754602687507, −3.990595085357607, −3.615044823634953, −2.881799681666652, −1.819696940244050, −1.009045309307607, 0,
1.009045309307607, 1.819696940244050, 2.881799681666652, 3.615044823634953, 3.990595085357607, 4.814754602687507, 5.246798714692609, 6.191339648462303, 6.882959217382957, 7.377333340246940, 7.833961561680686, 8.627637395954805, 8.817100133806418, 9.891258623986637, 10.09781624537977, 10.97682072108973, 11.65334453143358, 11.87086873304509, 12.20114361064168, 12.95459830776578, 13.65301710422316, 14.32650848925906, 14.74356554035622, 15.05417739668952, 15.74705533645298
