| L(s) = 1 | − 3-s − 5-s − 0.148·7-s + 9-s − 0.688·11-s + 0.0605·13-s + 15-s + 0.628·17-s − 0.223·19-s + 0.148·21-s + 1.29·23-s + 25-s − 27-s + 0.675·29-s − 4.47·31-s + 0.688·33-s + 0.148·35-s − 0.824·37-s − 0.0605·39-s − 2.75·41-s + 3.23·43-s − 45-s + 6.96·47-s − 6.97·49-s − 0.628·51-s − 6.23·53-s + 0.688·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.0561·7-s + 0.333·9-s − 0.207·11-s + 0.0167·13-s + 0.258·15-s + 0.152·17-s − 0.0512·19-s + 0.0324·21-s + 0.270·23-s + 0.200·25-s − 0.192·27-s + 0.125·29-s − 0.804·31-s + 0.119·33-s + 0.0251·35-s − 0.135·37-s − 0.00969·39-s − 0.429·41-s + 0.493·43-s − 0.149·45-s + 1.01·47-s − 0.996·49-s − 0.0879·51-s − 0.856·53-s + 0.0928·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 0.148T + 7T^{2} \) |
| 11 | \( 1 + 0.688T + 11T^{2} \) |
| 13 | \( 1 - 0.0605T + 13T^{2} \) |
| 17 | \( 1 - 0.628T + 17T^{2} \) |
| 19 | \( 1 + 0.223T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 - 0.675T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 0.824T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 4.01T + 97T^{2} \) |
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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
−8.056157887246743431826911292684, −7.24446157472570899069059848263, −6.70676856193545175513452707239, −5.76171861722187955065499547621, −5.16231336348146930962439361717, −4.28550402460427159833603255662, −3.52746957693013799406242982622, −2.48579144880744133215717027641, −1.25558269063534266064890542266, 0,
1.25558269063534266064890542266, 2.48579144880744133215717027641, 3.52746957693013799406242982622, 4.28550402460427159833603255662, 5.16231336348146930962439361717, 5.76171861722187955065499547621, 6.70676856193545175513452707239, 7.24446157472570899069059848263, 8.056157887246743431826911292684
