| L(s) = 1 | + 1.56·3-s − 5-s − 2.56·7-s − 0.561·9-s − 2·11-s − 3.56·13-s − 1.56·15-s + 1.43·17-s − 2·19-s − 4·21-s − 23-s + 25-s − 5.56·27-s − 8.12·29-s + 0.123·31-s − 3.12·33-s + 2.56·35-s + 0.561·37-s − 5.56·39-s + 4.12·41-s + 6.24·43-s + 0.561·45-s − 8.68·47-s − 0.438·49-s + 2.24·51-s + 8.56·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.447·5-s − 0.968·7-s − 0.187·9-s − 0.603·11-s − 0.987·13-s − 0.403·15-s + 0.348·17-s − 0.458·19-s − 0.872·21-s − 0.208·23-s + 0.200·25-s − 1.07·27-s − 1.50·29-s + 0.0221·31-s − 0.543·33-s + 0.432·35-s + 0.0923·37-s − 0.890·39-s + 0.643·41-s + 0.952·43-s + 0.0837·45-s − 1.26·47-s − 0.0626·49-s + 0.314·51-s + 1.17·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 0.123T + 31T^{2} \) |
| 37 | \( 1 - 0.561T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 0.876T + 61T^{2} \) |
| 67 | \( 1 + 7.43T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 + 0.876T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−9.527700492840389776085760445404, −8.918055005191489921962297743259, −7.87787094055281652610846232806, −7.41800561273076131734096444241, −6.26167348608412767415468072335, −5.25635349558868339763553177976, −3.99454410774820647075962821105, −3.11920822435374775943149076422, −2.27445453388986391326208563508, 0,
2.27445453388986391326208563508, 3.11920822435374775943149076422, 3.99454410774820647075962821105, 5.25635349558868339763553177976, 6.26167348608412767415468072335, 7.41800561273076131734096444241, 7.87787094055281652610846232806, 8.918055005191489921962297743259, 9.527700492840389776085760445404
