| L(s) = 1 | − 2·5-s + 6·11-s − 2·13-s + 17-s − 6·23-s − 25-s + 4·29-s − 4·31-s − 6·37-s + 6·41-s − 4·43-s + 12·47-s + 4·53-s − 12·55-s − 4·59-s − 2·61-s + 4·65-s − 4·67-s + 14·71-s − 10·73-s + 16·79-s − 8·83-s − 2·85-s − 10·89-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.80·11-s − 0.554·13-s + 0.242·17-s − 1.25·23-s − 1/5·25-s + 0.742·29-s − 0.718·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s + 0.549·53-s − 1.61·55-s − 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 1.66·71-s − 1.17·73-s + 1.80·79-s − 0.878·83-s − 0.216·85-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.93660699358702, −13.47639409019441, −12.44199468229298, −12.38172848402057, −11.87277480394535, −11.61728000087824, −10.90037897216131, −10.53118827366013, −9.675523218754293, −9.588640257667381, −8.770711224732333, −8.525079092551481, −7.797714916639748, −7.386821265835331, −6.918790082113826, −6.329864108509433, −5.856440169325563, −5.198862533697561, −4.469938581788447, −3.940884929153047, −3.783356216562371, −2.991132782680021, −2.200291002582212, −1.559756673614270, −0.8194579459846038, 0,
0.8194579459846038, 1.559756673614270, 2.200291002582212, 2.991132782680021, 3.783356216562371, 3.940884929153047, 4.469938581788447, 5.198862533697561, 5.856440169325563, 6.329864108509433, 6.918790082113826, 7.386821265835331, 7.797714916639748, 8.525079092551481, 8.770711224732333, 9.588640257667381, 9.675523218754293, 10.53118827366013, 10.90037897216131, 11.61728000087824, 11.87277480394535, 12.38172848402057, 12.44199468229298, 13.47639409019441, 13.93660699358702
