| L(s) = 1 | + 3-s − 2.24·5-s + 2.40·7-s + 9-s − 11-s − 13-s − 2.24·15-s + 0.711·17-s + 2.40·19-s + 2.40·21-s + 5.53·23-s + 0.0429·25-s + 27-s − 4.64·29-s − 0.559·31-s − 33-s − 5.39·35-s − 4.20·37-s − 39-s + 6.66·41-s − 2.71·43-s − 2.24·45-s + 5.35·47-s − 1.22·49-s + 0.711·51-s + 9.38·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.00·5-s + 0.908·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.579·15-s + 0.172·17-s + 0.551·19-s + 0.524·21-s + 1.15·23-s + 0.00859·25-s + 0.192·27-s − 0.863·29-s − 0.100·31-s − 0.174·33-s − 0.911·35-s − 0.690·37-s − 0.160·39-s + 1.04·41-s − 0.413·43-s − 0.334·45-s + 0.781·47-s − 0.175·49-s + 0.0996·51-s + 1.28·53-s + 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.222440798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222440798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 17 | \( 1 - 0.711T + 17T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 + 4.64T + 29T^{2} \) |
| 31 | \( 1 + 0.559T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 - 3.35T + 61T^{2} \) |
| 67 | \( 1 + 7.62T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 7.11T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
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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
−7.970303236984705336706649935810, −7.35642918048248602007146848622, −6.95953831942540750979414850338, −5.64819766113535223615983155680, −5.07010393053866752159800843650, −4.26116605857346343492384544411, −3.63147639661787255190637294927, −2.79587150851093110671750453355, −1.86459995994052622738307378980, −0.75030017997666982976146753102,
0.75030017997666982976146753102, 1.86459995994052622738307378980, 2.79587150851093110671750453355, 3.63147639661787255190637294927, 4.26116605857346343492384544411, 5.07010393053866752159800843650, 5.64819766113535223615983155680, 6.95953831942540750979414850338, 7.35642918048248602007146848622, 7.970303236984705336706649935810
