| L(s) = 1 | − 5-s + 7-s − 3·9-s − 4·13-s − 6·19-s + 23-s + 25-s + 6·29-s + 2·31-s − 35-s − 6·37-s + 6·41-s + 8·43-s + 3·45-s + 2·47-s + 49-s + 6·53-s + 8·59-s − 10·61-s − 3·63-s + 4·65-s − 8·67-s − 8·71-s − 10·73-s + 8·79-s + 9·81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 1.10·13-s − 1.37·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.169·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.291·47-s + 1/7·49-s + 0.824·53-s + 1.04·59-s − 1.28·61-s − 0.377·63-s + 0.496·65-s − 0.977·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.73276058465132, −14.39172483358394, −13.77890052078762, −13.25702188377851, −12.54362698170623, −12.06398932715502, −11.87305933469183, −11.09245005366973, −10.63417258408372, −10.29061326350951, −9.480233349510802, −8.799911684835691, −8.623567449153517, −7.893847141480790, −7.451062412916677, −6.833308417318602, −6.193282768670187, −5.644272827060450, −4.972485863384235, −4.437359656432810, −3.947463657171866, −2.932862206419922, −2.618328694099773, −1.864274444728661, −0.7894444527653480, 0,
0.7894444527653480, 1.864274444728661, 2.618328694099773, 2.932862206419922, 3.947463657171866, 4.437359656432810, 4.972485863384235, 5.644272827060450, 6.193282768670187, 6.833308417318602, 7.451062412916677, 7.893847141480790, 8.623567449153517, 8.799911684835691, 9.480233349510802, 10.29061326350951, 10.63417258408372, 11.09245005366973, 11.87305933469183, 12.06398932715502, 12.54362698170623, 13.25702188377851, 13.77890052078762, 14.39172483358394, 14.73276058465132
