| L(s) = 1 | − 5-s + 5·11-s + 2·13-s − 6·17-s − 2·19-s + 6·23-s − 4·25-s + 3·29-s + 5·31-s + 2·37-s − 8·41-s − 4·43-s + 4·47-s − 9·53-s − 5·55-s + 3·59-s − 12·61-s − 2·65-s + 2·67-s + 8·71-s + 14·73-s − 79-s − 17·83-s + 6·85-s − 18·89-s + 2·95-s − 3·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 0.328·37-s − 1.24·41-s − 0.609·43-s + 0.583·47-s − 1.23·53-s − 0.674·55-s + 0.390·59-s − 1.53·61-s − 0.248·65-s + 0.244·67-s + 0.949·71-s + 1.63·73-s − 0.112·79-s − 1.86·83-s + 0.650·85-s − 1.90·89-s + 0.205·95-s − 0.304·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−15.34836451182658, −15.23169866385875, −14.34253322217217, −13.88814726596658, −13.44140060476921, −12.78542461467251, −12.25246496474984, −11.61734167808948, −11.26260544238567, −10.82795591326938, −10.03119220788415, −9.411097095951107, −8.905385891570894, −8.437963894203811, −7.908288433889937, −6.917133351480515, −6.667285081294579, −6.204526697226877, −5.289782764305487, −4.493298045578080, −4.153663963065205, −3.450138091249182, −2.712255674380128, −1.769491303780047, −1.093419670973865, 0,
1.093419670973865, 1.769491303780047, 2.712255674380128, 3.450138091249182, 4.153663963065205, 4.493298045578080, 5.289782764305487, 6.204526697226877, 6.667285081294579, 6.917133351480515, 7.908288433889937, 8.437963894203811, 8.905385891570894, 9.411097095951107, 10.03119220788415, 10.82795591326938, 11.26260544238567, 11.61734167808948, 12.25246496474984, 12.78542461467251, 13.44140060476921, 13.88814726596658, 14.34253322217217, 15.23169866385875, 15.34836451182658
