| L(s) = 1 | − 2.16·3-s − 5-s + 7-s + 1.69·9-s + 4.88·11-s − 0.835·13-s + 2.16·15-s − 1.74·17-s − 0.528·19-s − 2.16·21-s − 5.66·23-s + 25-s + 2.82·27-s + 5.57·29-s − 10.1·31-s − 10.5·33-s − 35-s + 3.13·37-s + 1.81·39-s − 9.30·41-s − 43-s − 1.69·45-s + 12.3·47-s + 49-s + 3.78·51-s + 7.15·53-s − 4.88·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s − 0.447·5-s + 0.377·7-s + 0.565·9-s + 1.47·11-s − 0.231·13-s + 0.559·15-s − 0.423·17-s − 0.121·19-s − 0.472·21-s − 1.18·23-s + 0.200·25-s + 0.543·27-s + 1.03·29-s − 1.82·31-s − 1.84·33-s − 0.169·35-s + 0.515·37-s + 0.289·39-s − 1.45·41-s − 0.152·43-s − 0.252·45-s + 1.80·47-s + 0.142·49-s + 0.529·51-s + 0.982·53-s − 0.659·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 0.835T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 0.528T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 5.07T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 + 7.04T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−7.52984865420773102590526876528, −6.91391957668060000129725326408, −6.23851900837499537852624445880, −5.69612585527850655143296240211, −4.79721921075645575709610296648, −4.22911198325503315121035889457, −3.46406589491568328961640192870, −2.11366231838460113666240270228, −1.11443423402225187735254475061, 0,
1.11443423402225187735254475061, 2.11366231838460113666240270228, 3.46406589491568328961640192870, 4.22911198325503315121035889457, 4.79721921075645575709610296648, 5.69612585527850655143296240211, 6.23851900837499537852624445880, 6.91391957668060000129725326408, 7.52984865420773102590526876528
