| L(s) = 1 | − 2.41·3-s + 5-s + 2.82·9-s − 4.82·11-s + 2·13-s − 2.41·15-s + 3.65·17-s + 5.65·19-s − 8.41·23-s + 25-s + 0.414·27-s − 2.17·29-s − 4.82·31-s + 11.6·33-s − 5.65·37-s − 4.82·39-s + 0.171·41-s + 12.8·43-s + 2.82·45-s + 0.343·47-s − 8.82·51-s + 5.65·53-s − 4.82·55-s − 13.6·57-s − 4·59-s − 4.65·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s + 0.942·9-s − 1.45·11-s + 0.554·13-s − 0.623·15-s + 0.886·17-s + 1.29·19-s − 1.75·23-s + 0.200·25-s + 0.0797·27-s − 0.403·29-s − 0.867·31-s + 2.02·33-s − 0.929·37-s − 0.773·39-s + 0.0267·41-s + 1.96·43-s + 0.421·45-s + 0.0500·47-s − 1.23·51-s + 0.777·53-s − 0.651·55-s − 1.80·57-s − 0.520·59-s − 0.596·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8.41T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 0.171T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 4.65T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.840315981099739602337157609235, −7.77154923491413003868654168761, −7.25426298782425234681427305257, −5.99566180364328252561312775060, −5.67926474376942607471662334107, −5.09793864942923372879666655740, −3.92916244059385306929159040555, −2.72846026642977691943098186241, −1.37308574571468160271085307486, 0,
1.37308574571468160271085307486, 2.72846026642977691943098186241, 3.92916244059385306929159040555, 5.09793864942923372879666655740, 5.67926474376942607471662334107, 5.99566180364328252561312775060, 7.25426298782425234681427305257, 7.77154923491413003868654168761, 8.840315981099739602337157609235
