| L(s) = 1 | − 5-s + 5·7-s − 3·11-s − 4·13-s − 3·17-s + 2·23-s + 25-s − 4·29-s − 2·31-s − 5·35-s + 6·41-s − 9·43-s + 12·47-s + 18·49-s + 53-s + 3·55-s + 4·59-s − 11·61-s + 4·65-s − 3·67-s − 13·71-s − 14·73-s − 15·77-s + 4·79-s + 3·85-s + 12·89-s − 20·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.88·7-s − 0.904·11-s − 1.10·13-s − 0.727·17-s + 0.417·23-s + 1/5·25-s − 0.742·29-s − 0.359·31-s − 0.845·35-s + 0.937·41-s − 1.37·43-s + 1.75·47-s + 18/7·49-s + 0.137·53-s + 0.404·55-s + 0.520·59-s − 1.40·61-s + 0.496·65-s − 0.366·67-s − 1.54·71-s − 1.63·73-s − 1.70·77-s + 0.450·79-s + 0.325·85-s + 1.27·89-s − 2.09·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694875576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694875576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.08714503197756, −13.52651021847169, −13.10272840969827, −12.40132782315993, −12.01927657271740, −11.43840449114885, −11.14366392738140, −10.53433218694451, −10.26472185819201, −9.390084453310015, −8.767300344056224, −8.542741439332286, −7.638967808029795, −7.556052260224856, −7.178234257245567, −6.191578352046823, −5.494080086672144, −5.108538356464627, −4.500854078919424, −4.283712972795809, −3.297053033510225, −2.531833011327083, −2.077695584701603, −1.384442941282537, −0.4218634177859211,
0.4218634177859211, 1.384442941282537, 2.077695584701603, 2.531833011327083, 3.297053033510225, 4.283712972795809, 4.500854078919424, 5.108538356464627, 5.494080086672144, 6.191578352046823, 7.178234257245567, 7.556052260224856, 7.638967808029795, 8.542741439332286, 8.767300344056224, 9.390084453310015, 10.26472185819201, 10.53433218694451, 11.14366392738140, 11.43840449114885, 12.01927657271740, 12.40132782315993, 13.10272840969827, 13.52651021847169, 14.08714503197756
