| L(s) = 1 | − 2.37·3-s + 0.716·5-s − 7-s + 2.64·9-s + 1.73·11-s + 1.39·13-s − 1.70·15-s + 17-s − 4.12·19-s + 2.37·21-s − 1.18·23-s − 4.48·25-s + 0.849·27-s + 6.51·29-s − 6.37·31-s − 4.12·33-s − 0.716·35-s − 8.59·37-s − 3.30·39-s + 9.04·41-s − 9.08·43-s + 1.89·45-s + 3.67·47-s + 49-s − 2.37·51-s + 9.86·53-s + 1.24·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 0.320·5-s − 0.377·7-s + 0.880·9-s + 0.523·11-s + 0.385·13-s − 0.439·15-s + 0.242·17-s − 0.946·19-s + 0.518·21-s − 0.248·23-s − 0.897·25-s + 0.163·27-s + 1.21·29-s − 1.14·31-s − 0.718·33-s − 0.121·35-s − 1.41·37-s − 0.528·39-s + 1.41·41-s − 1.38·43-s + 0.282·45-s + 0.535·47-s + 0.142·49-s − 0.332·51-s + 1.35·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.716T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 1.39T + 13T^{2} \) |
| 19 | \( 1 + 4.12T + 19T^{2} \) |
| 23 | \( 1 + 1.18T + 23T^{2} \) |
| 29 | \( 1 - 6.51T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + 8.59T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 + 2.84T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.207488111037579585719499034365, −6.95184948192511354161659529570, −6.67268460044080262679739615545, −5.74908511425815616164756270994, −5.45512161796180508006142189738, −4.34980540183823317500781684766, −3.66750024114580001741295949431, −2.34560253224922895190592095453, −1.20437185677027119307468636464, 0,
1.20437185677027119307468636464, 2.34560253224922895190592095453, 3.66750024114580001741295949431, 4.34980540183823317500781684766, 5.45512161796180508006142189738, 5.74908511425815616164756270994, 6.67268460044080262679739615545, 6.95184948192511354161659529570, 8.207488111037579585719499034365
