| L(s) = 1 | − 5-s + 7-s + 3·11-s + 5.29·13-s + 5.29·17-s + 5.29·19-s + 5.29·23-s − 4·25-s + 6·29-s + 7·31-s − 35-s − 5.29·37-s + 5.29·41-s − 10.5·43-s − 6·49-s + 9·53-s − 3·55-s − 4·59-s − 5.29·65-s − 15.8·71-s − 3·73-s + 3·77-s − 4·79-s + 7·83-s − 5.29·85-s − 10.5·89-s + 5.29·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s + 1.46·13-s + 1.28·17-s + 1.21·19-s + 1.10·23-s − 0.800·25-s + 1.11·29-s + 1.25·31-s − 0.169·35-s − 0.869·37-s + 0.826·41-s − 1.61·43-s − 0.857·49-s + 1.23·53-s − 0.404·55-s − 0.520·59-s − 0.656·65-s − 1.88·71-s − 0.351·73-s + 0.341·77-s − 0.450·79-s + 0.768·83-s − 0.573·85-s − 1.12·89-s + 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.682186710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682186710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.020381197661248698718192528167, −7.30432295425651012562502237548, −6.56178477552353195606235272832, −5.88268255963803482179124709230, −5.12098813437433552223115508905, −4.31403324474316440739449150089, −3.49385546113585354226210487085, −3.01898593901985181909277463273, −1.47536598950536411222855825324, −0.976856393918304537559425562636,
0.976856393918304537559425562636, 1.47536598950536411222855825324, 3.01898593901985181909277463273, 3.49385546113585354226210487085, 4.31403324474316440739449150089, 5.12098813437433552223115508905, 5.88268255963803482179124709230, 6.56178477552353195606235272832, 7.30432295425651012562502237548, 8.020381197661248698718192528167
