| L(s) = 1 | + 2·5-s + 2·13-s − 17-s + 6·19-s + 8·23-s − 25-s + 4·29-s + 10·37-s + 10·41-s + 8·43-s + 8·47-s − 6·53-s + 4·59-s + 8·61-s + 4·65-s + 4·67-s − 8·71-s − 4·73-s − 2·85-s + 10·89-s + 12·95-s + 101-s + 103-s + 107-s + 109-s + 113-s + 16·115-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.554·13-s − 0.242·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s + 0.742·29-s + 1.64·37-s + 1.56·41-s + 1.21·43-s + 1.16·47-s − 0.824·53-s + 0.520·59-s + 1.02·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s − 0.468·73-s − 0.216·85-s + 1.05·89-s + 1.23·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.49·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.855967198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.855967198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.51400559098994, −13.08525109318997, −12.81578588101649, −12.13419552897130, −11.48071302816492, −11.23624198656535, −10.60381266039622, −10.20494973818066, −9.512651254849715, −9.206223288421427, −8.919251665655955, −8.047018641487840, −7.645703509502011, −7.099713414172506, −6.507638527674114, −5.968502335293318, −5.578036699561367, −5.047273606416579, −4.369693822570431, −3.866793036851339, −2.948139348550544, −2.710684250635496, −1.958807491230646, −1.010691201227427, −0.8548435964715003,
0.8548435964715003, 1.010691201227427, 1.958807491230646, 2.710684250635496, 2.948139348550544, 3.866793036851339, 4.369693822570431, 5.047273606416579, 5.578036699561367, 5.968502335293318, 6.507638527674114, 7.099713414172506, 7.645703509502011, 8.047018641487840, 8.919251665655955, 9.206223288421427, 9.512651254849715, 10.20494973818066, 10.60381266039622, 11.23624198656535, 11.48071302816492, 12.13419552897130, 12.81578588101649, 13.08525109318997, 13.51400559098994
