L(s) = 1 | + 5-s − 7-s + 4·11-s − 6·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s + 10·29-s + 4·31-s − 35-s + 6·37-s − 6·41-s − 4·43-s + 12·47-s + 49-s + 6·53-s + 4·55-s + 4·59-s + 2·61-s − 6·65-s − 4·67-s − 2·73-s − 4·77-s − 8·79-s − 12·83-s + 6·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s − 0.169·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.744·65-s − 0.488·67-s − 0.234·73-s − 0.455·77-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.695113820\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.695113820\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.46162753962275, −15.07621745523092, −14.43460553089607, −14.24436811328337, −13.48930428243604, −12.84672453328966, −12.27857843680070, −11.99411955803413, −11.35128032984983, −10.46465035871655, −9.958319982266145, −9.752541721490935, −8.831227430382729, −8.578278521331939, −7.582546928097106, −7.030154273461644, −6.581669952092283, −5.899845156185951, −5.164196607802828, −4.601387220435844, −3.906144723258304, −2.884291560862794, −2.614044680951736, −1.425103392013025, −0.7286530597117739,
0.7286530597117739, 1.425103392013025, 2.614044680951736, 2.884291560862794, 3.906144723258304, 4.601387220435844, 5.164196607802828, 5.899845156185951, 6.581669952092283, 7.030154273461644, 7.582546928097106, 8.578278521331939, 8.831227430382729, 9.752541721490935, 9.958319982266145, 10.46465035871655, 11.35128032984983, 11.99411955803413, 12.27857843680070, 12.84672453328966, 13.48930428243604, 14.24436811328337, 14.43460553089607, 15.07621745523092, 15.46162753962275