| L(s) = 1 | + 3-s + 2·5-s − 2·7-s − 2·9-s − 11-s − 4·13-s + 2·15-s + 2·17-s − 2·21-s + 6·23-s − 25-s − 5·27-s + 2·29-s + 10·31-s − 33-s − 4·35-s − 10·37-s − 4·39-s + 9·41-s + 4·43-s − 4·45-s + 6·47-s − 3·49-s + 2·51-s − 2·55-s + 15·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.485·17-s − 0.436·21-s + 1.25·23-s − 1/5·25-s − 0.962·27-s + 0.371·29-s + 1.79·31-s − 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.640·39-s + 1.40·41-s + 0.609·43-s − 0.596·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.269·55-s + 1.95·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.436192263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.436192263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.89772825374670, −13.40251378938740, −12.93392628449116, −12.50489370231195, −11.85801347263062, −11.53615055934921, −10.66930359509484, −10.18382590027566, −9.950176634069589, −9.256859219124189, −9.013471925579120, −8.403149251123764, −7.815464518008112, −7.219794472233520, −6.809818580025266, −6.027546152759814, −5.740585286730990, −5.103649910670145, −4.571883519573859, −3.736214767898632, −3.067151653021794, −2.618595982723420, −2.299173935536901, −1.313844028357106, −0.4752540979405293,
0.4752540979405293, 1.313844028357106, 2.299173935536901, 2.618595982723420, 3.067151653021794, 3.736214767898632, 4.571883519573859, 5.103649910670145, 5.740585286730990, 6.027546152759814, 6.809818580025266, 7.219794472233520, 7.815464518008112, 8.403149251123764, 9.013471925579120, 9.256859219124189, 9.950176634069589, 10.18382590027566, 10.66930359509484, 11.53615055934921, 11.85801347263062, 12.50489370231195, 12.93392628449116, 13.40251378938740, 13.89772825374670
