| L(s) = 1 | + 3-s + 1.10·5-s + 2.52·7-s + 9-s − 0.813·11-s − 5.10·13-s + 1.10·15-s + 3.39·17-s − 3.10·19-s + 2.52·21-s − 0.897·23-s − 3.78·25-s + 27-s − 4.44·29-s − 8.96·31-s − 0.813·33-s + 2.78·35-s − 2.08·37-s − 5.10·39-s + 41-s − 9.07·43-s + 1.10·45-s − 0.235·47-s − 0.627·49-s + 3.39·51-s − 13.8·53-s − 0.897·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.493·5-s + 0.954·7-s + 0.333·9-s − 0.245·11-s − 1.41·13-s + 0.284·15-s + 0.822·17-s − 0.711·19-s + 0.550·21-s − 0.187·23-s − 0.756·25-s + 0.192·27-s − 0.824·29-s − 1.61·31-s − 0.141·33-s + 0.470·35-s − 0.342·37-s − 0.817·39-s + 0.156·41-s − 1.38·43-s + 0.164·45-s − 0.0343·47-s − 0.0896·49-s + 0.474·51-s − 1.90·53-s − 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 0.235T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 1.91T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.72T + 97T^{2} \) |
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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
−7.72343238222036682900682114163, −6.99284142150821604463401344739, −6.05399442810948887351669922622, −5.25791199316510897323863016118, −4.80959221299238608162905897029, −3.87850051495049421789602486543, −3.04656985347670052195096756849, −2.02092593033102794770452681570, −1.69734575821843916572800678515, 0,
1.69734575821843916572800678515, 2.02092593033102794770452681570, 3.04656985347670052195096756849, 3.87850051495049421789602486543, 4.80959221299238608162905897029, 5.25791199316510897323863016118, 6.05399442810948887351669922622, 6.99284142150821604463401344739, 7.72343238222036682900682114163
