| L(s) = 1 | + 1.17·3-s − 0.630·7-s − 1.63·9-s − 0.539·11-s + 13-s − 3.07·17-s + 0.879·19-s − 0.738·21-s − 2.58·23-s − 5.41·27-s − 2.29·29-s − 7.95·31-s − 0.630·33-s − 4.78·37-s + 1.17·39-s + 3.41·41-s + 3.17·43-s + 8.94·47-s − 6.60·49-s − 3.60·51-s + 0.496·53-s + 1.02·57-s − 8.29·59-s + 4.04·61-s + 1.02·63-s − 3.36·67-s − 3.02·69-s + ⋯ |
| L(s) = 1 | + 0.675·3-s − 0.238·7-s − 0.543·9-s − 0.162·11-s + 0.277·13-s − 0.746·17-s + 0.201·19-s − 0.161·21-s − 0.539·23-s − 1.04·27-s − 0.425·29-s − 1.42·31-s − 0.109·33-s − 0.787·37-s + 0.187·39-s + 0.533·41-s + 0.483·43-s + 1.30·47-s − 0.943·49-s − 0.504·51-s + 0.0682·53-s + 0.136·57-s − 1.08·59-s + 0.518·61-s + 0.129·63-s − 0.411·67-s − 0.364·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 0.539T + 11T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 - 0.879T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 0.496T + 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 3.36T + 67T^{2} \) |
| 71 | \( 1 + 9.06T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 1.02T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−8.701648119092967067837828985741, −7.73885903047633846715170375038, −7.16064906000284519192217380428, −6.08679876292915570015421552531, −5.51216170551403283654101000075, −4.35676499414583572282300717183, −3.53489843593616309985182746781, −2.70090067063046327925638638988, −1.75440020857637769243245589445, 0,
1.75440020857637769243245589445, 2.70090067063046327925638638988, 3.53489843593616309985182746781, 4.35676499414583572282300717183, 5.51216170551403283654101000075, 6.08679876292915570015421552531, 7.16064906000284519192217380428, 7.73885903047633846715170375038, 8.701648119092967067837828985741
