| L(s) = 1 | + 2·3-s + 5-s + 9-s − 4·11-s + 6·13-s + 2·15-s − 2·17-s − 8·19-s − 6·23-s + 25-s − 4·27-s − 2·29-s − 4·31-s − 8·33-s + 2·37-s + 12·39-s + 10·41-s − 2·43-s + 45-s + 2·47-s − 4·51-s + 2·53-s − 4·55-s − 16·57-s − 2·61-s + 6·65-s − 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.516·15-s − 0.485·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.39·33-s + 0.328·37-s + 1.92·39-s + 1.56·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s − 0.560·51-s + 0.274·53-s − 0.539·55-s − 2.11·57-s − 0.256·61-s + 0.744·65-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.83938537083331631256637178062, −6.85425115459008901949951264155, −5.95399035311661128750600979066, −5.70422636800381836064399540615, −4.36257795409630888270702034421, −3.93124254095635158199786653018, −2.95678195034229114542622482450, −2.30681786687542446203602074113, −1.63017756465829949468805905873, 0,
1.63017756465829949468805905873, 2.30681786687542446203602074113, 2.95678195034229114542622482450, 3.93124254095635158199786653018, 4.36257795409630888270702034421, 5.70422636800381836064399540615, 5.95399035311661128750600979066, 6.85425115459008901949951264155, 7.83938537083331631256637178062
