| L(s) = 1 | − 2·5-s + 2·13-s + 2·17-s − 4·19-s − 25-s + 6·29-s − 6·37-s − 6·41-s + 8·43-s − 8·47-s + 6·53-s − 12·59-s + 10·61-s − 4·65-s + 16·67-s − 8·71-s + 6·73-s − 8·79-s − 12·83-s − 4·85-s − 14·89-s + 8·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.95·67-s − 0.949·71-s + 0.702·73-s − 0.900·79-s − 1.31·83-s − 0.433·85-s − 1.48·89-s + 0.820·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375155043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375155043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.38938046297700, −14.71230250954057, −14.10394868106012, −13.77931546858861, −12.89569055444601, −12.59128964593916, −12.00088198921025, −11.38736227927302, −11.10885582684856, −10.19491723935298, −10.06306059922957, −9.084383236446752, −8.449978519190837, −8.264211040979859, −7.477005956285677, −6.945812854687462, −6.318122233701774, −5.701593526215243, −4.931574156862217, −4.321512542781922, −3.712176570803730, −3.173197323921602, −2.289412450178215, −1.419014847543545, −0.4705511050177780,
0.4705511050177780, 1.419014847543545, 2.289412450178215, 3.173197323921602, 3.712176570803730, 4.321512542781922, 4.931574156862217, 5.701593526215243, 6.318122233701774, 6.945812854687462, 7.477005956285677, 8.264211040979859, 8.449978519190837, 9.084383236446752, 10.06306059922957, 10.19491723935298, 11.10885582684856, 11.38736227927302, 12.00088198921025, 12.59128964593916, 12.89569055444601, 13.77931546858861, 14.10394868106012, 14.71230250954057, 15.38938046297700
