| L(s) = 1 | + 1.55·3-s + 2·5-s − 4.48·7-s − 0.591·9-s + 1.62·11-s + 3.38·13-s + 3.10·15-s − 5.30·17-s − 6.97·19-s − 6.96·21-s − 0.132·23-s − 25-s − 5.57·27-s − 7.86·29-s − 10.5·31-s + 2.52·33-s − 8.97·35-s + 1.72·37-s + 5.25·39-s + 8.40·41-s + 6.76·43-s − 1.18·45-s − 0.208·47-s + 13.1·49-s − 8.23·51-s + 8.68·53-s + 3.25·55-s + ⋯ |
| L(s) = 1 | + 0.896·3-s + 0.894·5-s − 1.69·7-s − 0.197·9-s + 0.489·11-s + 0.938·13-s + 0.801·15-s − 1.28·17-s − 1.59·19-s − 1.51·21-s − 0.0276·23-s − 0.200·25-s − 1.07·27-s − 1.46·29-s − 1.90·31-s + 0.439·33-s − 1.51·35-s + 0.282·37-s + 0.840·39-s + 1.31·41-s + 1.03·43-s − 0.176·45-s − 0.0304·47-s + 1.87·49-s − 1.15·51-s + 1.19·53-s + 0.438·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 0.132T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 1.72T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 0.208T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 + 7.51T + 67T^{2} \) |
| 71 | \( 1 + 2.06T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 4.21T + 97T^{2} \) |
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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
−8.897929495916341751290784009813, −7.76056936989097604305318098105, −6.80745454434071098347125055968, −6.10936801557641279425744316447, −5.76550243797192046972350338624, −4.08893808136322460145735014490, −3.63204355291565254358841339111, −2.55623973214421916596031780439, −1.91682561110098259869604092218, 0,
1.91682561110098259869604092218, 2.55623973214421916596031780439, 3.63204355291565254358841339111, 4.08893808136322460145735014490, 5.76550243797192046972350338624, 6.10936801557641279425744316447, 6.80745454434071098347125055968, 7.76056936989097604305318098105, 8.897929495916341751290784009813
