| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s + 10·43-s − 45-s + 6·47-s − 3·49-s + 12·51-s − 6·53-s + 8·57-s + 12·59-s − 2·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.195786377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.195786377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.77497486915745, −15.25582452520766, −14.67238707740285, −14.28788405784851, −13.89528445451996, −13.23152846458340, −12.68258445955798, −11.82508069842703, −11.60807922718766, −10.83394756213898, −10.23365445716969, −9.535100593390327, −8.942688621097727, −8.483197285337311, −7.847760534236091, −7.546829682891877, −6.851649162780267, −5.832822396766763, −5.283430344596769, −4.543011599389774, −3.717085824744381, −3.199029104729932, −2.628545365498039, −1.564333829812903, −0.8926823574174181,
0.8926823574174181, 1.564333829812903, 2.628545365498039, 3.199029104729932, 3.717085824744381, 4.543011599389774, 5.283430344596769, 5.832822396766763, 6.851649162780267, 7.546829682891877, 7.847760534236091, 8.483197285337311, 8.942688621097727, 9.535100593390327, 10.23365445716969, 10.83394756213898, 11.60807922718766, 11.82508069842703, 12.68258445955798, 13.23152846458340, 13.89528445451996, 14.28788405784851, 14.67238707740285, 15.25582452520766, 15.77497486915745
