| L(s) = 1 | − 3-s + 0.461·5-s + 0.732·7-s + 9-s + 0.317·11-s + 1.73·13-s − 0.461·15-s − 1.85·17-s − 0.400·19-s − 0.732·21-s − 23-s − 4.78·25-s − 27-s + 29-s − 2.76·31-s − 0.317·33-s + 0.338·35-s + 6.86·37-s − 1.73·39-s + 2.88·41-s + 7.54·43-s + 0.461·45-s − 11.3·47-s − 6.46·49-s + 1.85·51-s − 10.8·53-s + 0.146·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.206·5-s + 0.276·7-s + 0.333·9-s + 0.0955·11-s + 0.480·13-s − 0.119·15-s − 0.450·17-s − 0.0919·19-s − 0.159·21-s − 0.208·23-s − 0.957·25-s − 0.192·27-s + 0.185·29-s − 0.496·31-s − 0.0551·33-s + 0.0571·35-s + 1.12·37-s − 0.277·39-s + 0.450·41-s + 1.15·43-s + 0.0688·45-s − 1.66·47-s − 0.923·49-s + 0.260·51-s − 1.48·53-s + 0.0197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 0.461T + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 0.317T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 0.400T + 19T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 6.86T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 + 0.411T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.61871782409178494809467470684, −6.55661156504915414727657478300, −6.18467423373851886055354599101, −5.47067709502495375422485714782, −4.65415310107979136225911915983, −4.07361393800499387324790832171, −3.12659056681963479877969488030, −2.08031465828343288149424535739, −1.27063393933418541834367086869, 0,
1.27063393933418541834367086869, 2.08031465828343288149424535739, 3.12659056681963479877969488030, 4.07361393800499387324790832171, 4.65415310107979136225911915983, 5.47067709502495375422485714782, 6.18467423373851886055354599101, 6.55661156504915414727657478300, 7.61871782409178494809467470684
