| L(s) = 1 | + 3-s − 5-s − 0.585·7-s + 9-s − 1.41·11-s + 5.41·13-s − 15-s − 1.17·17-s − 19-s − 0.585·21-s − 7.65·23-s + 25-s + 27-s − 9.07·29-s − 6.48·31-s − 1.41·33-s + 0.585·35-s + 11.0·37-s + 5.41·39-s − 7.41·41-s − 0.585·43-s − 45-s − 0.343·47-s − 6.65·49-s − 1.17·51-s + 4·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.221·7-s + 0.333·9-s − 0.426·11-s + 1.50·13-s − 0.258·15-s − 0.284·17-s − 0.229·19-s − 0.127·21-s − 1.59·23-s + 0.200·25-s + 0.192·27-s − 1.68·29-s − 1.16·31-s − 0.246·33-s + 0.0990·35-s + 1.82·37-s + 0.866·39-s − 1.15·41-s − 0.0893·43-s − 0.149·45-s − 0.0500·47-s − 0.950·49-s − 0.164·51-s + 0.549·53-s + 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 9.07T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.585T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−7.912672064533223106069533593596, −7.52255334983447452150479148029, −6.42369365317686094962313402552, −5.93716324673394462646985309396, −4.91565969754336242605934988608, −3.80542225875426960661118457347, −3.65222852485894003003971265276, −2.42347099106709586935239125445, −1.51898054597079972778560392322, 0,
1.51898054597079972778560392322, 2.42347099106709586935239125445, 3.65222852485894003003971265276, 3.80542225875426960661118457347, 4.91565969754336242605934988608, 5.93716324673394462646985309396, 6.42369365317686094962313402552, 7.52255334983447452150479148029, 7.912672064533223106069533593596
