| L(s) = 1 | − 3-s − 2.84·5-s − 4.19·7-s + 9-s − 4.69·11-s − 0.144·13-s + 2.84·15-s + 2.14·17-s + 2.69·19-s + 4.19·21-s + 4.69·23-s + 3.09·25-s − 27-s − 8.98·29-s − 4.50·31-s + 4.69·33-s + 11.9·35-s + 7.39·37-s + 0.144·39-s + 12.5·41-s − 6.60·43-s − 2.84·45-s + 6.47·47-s + 10.5·49-s − 2.14·51-s + 8.22·53-s + 13.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.27·5-s − 1.58·7-s + 0.333·9-s − 1.41·11-s − 0.0399·13-s + 0.734·15-s + 0.520·17-s + 0.619·19-s + 0.914·21-s + 0.979·23-s + 0.619·25-s − 0.192·27-s − 1.66·29-s − 0.809·31-s + 0.817·33-s + 2.01·35-s + 1.21·37-s + 0.0230·39-s + 1.95·41-s − 1.00·43-s − 0.424·45-s + 0.945·47-s + 1.50·49-s − 0.300·51-s + 1.12·53-s + 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 + 0.144T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 + 8.98T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 9.85T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 3.37T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.35968321199913475555867070511, −7.07548690059560620298486427422, −5.89965601709177010814999061344, −5.59679827430751157844815436232, −4.64407472137369257465940749424, −3.78843120829777776433268160995, −3.25114028696563933442572696232, −2.46607946211698665377429602429, −0.78566999108452655594840526518, 0,
0.78566999108452655594840526518, 2.46607946211698665377429602429, 3.25114028696563933442572696232, 3.78843120829777776433268160995, 4.64407472137369257465940749424, 5.59679827430751157844815436232, 5.89965601709177010814999061344, 7.07548690059560620298486427422, 7.35968321199913475555867070511
