| L(s) = 1 | + 1.73·3-s − 2.65·5-s − 3.64·7-s + 0.0151·9-s + 11-s + 5.02·13-s − 4.60·15-s − 1.30·17-s + 6.63·19-s − 6.33·21-s − 1.29·23-s + 2.03·25-s − 5.18·27-s − 3.04·29-s + 5.27·31-s + 1.73·33-s + 9.66·35-s + 7.89·37-s + 8.71·39-s − 7.88·41-s + 2.13·43-s − 0.0400·45-s + 2.44·47-s + 6.29·49-s − 2.26·51-s + 5.73·53-s − 2.65·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 1.18·5-s − 1.37·7-s + 0.00504·9-s + 0.301·11-s + 1.39·13-s − 1.18·15-s − 0.316·17-s + 1.52·19-s − 1.38·21-s − 0.269·23-s + 0.406·25-s − 0.997·27-s − 0.565·29-s + 0.947·31-s + 0.302·33-s + 1.63·35-s + 1.29·37-s + 1.39·39-s − 1.23·41-s + 0.325·43-s − 0.00597·45-s + 0.356·47-s + 0.899·49-s − 0.316·51-s + 0.787·53-s − 0.357·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 + 3.04T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 + 7.88T + 41T^{2} \) |
| 43 | \( 1 - 2.13T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 3.11T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 + 3.91T + 97T^{2} \) |
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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
−7.70942381612205412595542156608, −7.28428997529072960475652505035, −6.27887724465692330355984804103, −5.83172698704550982730420557649, −4.49598336379722939415391344986, −3.72264264908525562937109489851, −3.32279424405943324278579728220, −2.72061101180930026994423645967, −1.27425928659068079655510136632, 0,
1.27425928659068079655510136632, 2.72061101180930026994423645967, 3.32279424405943324278579728220, 3.72264264908525562937109489851, 4.49598336379722939415391344986, 5.83172698704550982730420557649, 6.27887724465692330355984804103, 7.28428997529072960475652505035, 7.70942381612205412595542156608
