L(s) = 1 | − 3·7-s + 3·11-s − 13-s − 7·17-s − 8·19-s + 4·23-s − 3·29-s + 11·31-s + 2·41-s − 8·43-s − 9·47-s + 2·49-s + 9·53-s + 9·59-s − 61-s + 5·67-s − 12·73-s − 9·77-s + 8·79-s + 9·83-s − 12·89-s + 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 0.904·11-s − 0.277·13-s − 1.69·17-s − 1.83·19-s + 0.834·23-s − 0.557·29-s + 1.97·31-s + 0.312·41-s − 1.21·43-s − 1.31·47-s + 2/7·49-s + 1.23·53-s + 1.17·59-s − 0.128·61-s + 0.610·67-s − 1.40·73-s − 1.02·77-s + 0.900·79-s + 0.987·83-s − 1.27·89-s + 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
\(L(1)\) |
\(\approx\) |
\(0.6076293439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6076293439\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.08110760405101, −12.93630407390339, −12.03579885133462, −11.86907889978807, −11.24322531144326, −10.69680270964558, −10.38573710141513, −9.650191074882232, −9.436215704150976, −8.849315152739613, −8.405442332459708, −8.066509726785856, −6.990853617687506, −6.731651060865496, −6.551469275410511, −6.030585351663403, −5.244399735140878, −4.640871258030927, −4.147448215198136, −3.788860578718898, −2.965314455386035, −2.523596498130865, −1.939721093695742, −1.138942595707813, −0.2302445818779757,
0.2302445818779757, 1.138942595707813, 1.939721093695742, 2.523596498130865, 2.965314455386035, 3.788860578718898, 4.147448215198136, 4.640871258030927, 5.244399735140878, 6.030585351663403, 6.551469275410511, 6.731651060865496, 6.990853617687506, 8.066509726785856, 8.405442332459708, 8.849315152739613, 9.436215704150976, 9.650191074882232, 10.38573710141513, 10.69680270964558, 11.24322531144326, 11.86907889978807, 12.03579885133462, 12.93630407390339, 13.08110760405101