| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 5.12·7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 3.12·13-s + 5.12·14-s + 15-s + 16-s − 17-s − 18-s − 5.12·19-s − 20-s + 5.12·21-s + 22-s + 5.12·23-s + 24-s + 25-s + 3.12·26-s − 27-s − 5.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.93·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.866·13-s + 1.36·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.17·19-s − 0.223·20-s + 1.11·21-s + 0.213·22-s + 1.06·23-s + 0.204·24-s + 0.200·25-s + 0.612·26-s − 0.192·27-s − 0.968·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 0.876T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.72579528193371846491705213349, −6.93941952073029132568959007777, −6.52161677746673874930792949851, −5.95847921420948554713859963582, −4.85278165459810449003758763802, −4.09793630490715019314108975147, −2.97532646919772594957301820161, −2.53261331735217934787494582682, −0.849150571873832115931609819761, 0,
0.849150571873832115931609819761, 2.53261331735217934787494582682, 2.97532646919772594957301820161, 4.09793630490715019314108975147, 4.85278165459810449003758763802, 5.95847921420948554713859963582, 6.52161677746673874930792949851, 6.93941952073029132568959007777, 7.72579528193371846491705213349
