| L(s) = 1 | + 2.87·3-s − 5-s + 5.29·9-s − 1.46·11-s − 2.22·13-s − 2.87·15-s − 7.26·17-s − 5.48·19-s − 2.51·23-s + 25-s + 6.60·27-s − 7.12·29-s − 6.51·31-s − 4.22·33-s + 6.90·37-s − 6.39·39-s + 11.3·41-s − 3.31·43-s − 5.29·45-s − 8.36·47-s − 20.9·51-s − 7.00·53-s + 1.46·55-s − 15.8·57-s + 9.07·59-s − 11.2·61-s + 2.22·65-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.447·5-s + 1.76·9-s − 0.441·11-s − 0.616·13-s − 0.743·15-s − 1.76·17-s − 1.25·19-s − 0.524·23-s + 0.200·25-s + 1.27·27-s − 1.32·29-s − 1.17·31-s − 0.734·33-s + 1.13·37-s − 1.02·39-s + 1.76·41-s − 0.505·43-s − 0.789·45-s − 1.22·47-s − 2.93·51-s − 0.962·53-s + 0.197·55-s − 2.09·57-s + 1.18·59-s − 1.43·61-s + 0.275·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 7.26T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 6.51T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.31T + 43T^{2} \) |
| 47 | \( 1 + 8.36T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−8.099992612805886704907524597409, −7.60140026752400967561298085016, −6.91701937561758439111637411029, −6.00527767061376915983765883787, −4.69501658284931436752912629323, −4.17398054714658365197633070452, −3.38144591894135661928402628872, −2.37212688521159150142968489047, −1.97413424286168116531305493673, 0,
1.97413424286168116531305493673, 2.37212688521159150142968489047, 3.38144591894135661928402628872, 4.17398054714658365197633070452, 4.69501658284931436752912629323, 6.00527767061376915983765883787, 6.91701937561758439111637411029, 7.60140026752400967561298085016, 8.099992612805886704907524597409
