L(s) = 1 | + 13-s + 6·17-s + 8·19-s − 6·23-s − 2·29-s + 6·31-s − 2·37-s − 10·41-s − 4·43-s + 4·47-s − 7·49-s + 4·53-s − 4·59-s + 2·61-s − 10·67-s + 12·73-s + 4·83-s − 14·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.277·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s − 0.371·29-s + 1.07·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s − 49-s + 0.549·53-s − 0.520·59-s + 0.256·61-s − 1.22·67-s + 1.40·73-s + 0.439·83-s − 1.48·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.413889408\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.413889408\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.35914454317467, −12.46526304029497, −12.06379376288141, −11.89190603548357, −11.35823014026772, −10.73715843981834, −10.13288424417387, −9.851368340210790, −9.532421275973703, −8.792787662232087, −8.278654921500515, −7.826515632409042, −7.494638842399321, −6.846208958440747, −6.305389076915281, −5.793289667784531, −5.171652480415210, −5.030633209804864, −4.025751828466724, −3.661135674513543, −3.087180765922624, −2.615070955922700, −1.597523438977111, −1.325237288483687, −0.4560655539652381,
0.4560655539652381, 1.325237288483687, 1.597523438977111, 2.615070955922700, 3.087180765922624, 3.661135674513543, 4.025751828466724, 5.030633209804864, 5.171652480415210, 5.793289667784531, 6.305389076915281, 6.846208958440747, 7.494638842399321, 7.826515632409042, 8.278654921500515, 8.792787662232087, 9.532421275973703, 9.851368340210790, 10.13288424417387, 10.73715843981834, 11.35823014026772, 11.89190603548357, 12.06379376288141, 12.46526304029497, 13.35914454317467