| L(s) = 1 | + 3.13·3-s + 0.555·5-s + 7-s + 6.81·9-s + 3.14·11-s + 4.47·13-s + 1.74·15-s − 0.980·17-s + 7.44·19-s + 3.13·21-s − 1.25·23-s − 4.69·25-s + 11.9·27-s + 4.48·29-s − 4.43·31-s + 9.86·33-s + 0.555·35-s − 0.912·37-s + 14.0·39-s + 1.21·41-s + 1.36·43-s + 3.79·45-s − 9.97·47-s + 49-s − 3.07·51-s − 11.4·53-s + 1.75·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 0.248·5-s + 0.377·7-s + 2.27·9-s + 0.949·11-s + 1.24·13-s + 0.449·15-s − 0.237·17-s + 1.70·19-s + 0.683·21-s − 0.262·23-s − 0.938·25-s + 2.30·27-s + 0.833·29-s − 0.795·31-s + 1.71·33-s + 0.0939·35-s − 0.149·37-s + 2.24·39-s + 0.189·41-s + 0.208·43-s + 0.565·45-s − 1.45·47-s + 0.142·49-s − 0.430·51-s − 1.56·53-s + 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.581546184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.581546184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 - 0.555T + 5T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.980T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 0.912T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 + 0.934T + 71T^{2} \) |
| 73 | \( 1 - 0.710T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−7.987400731958146934547129163785, −7.50307494821165795279248770826, −6.67489344114888901496490848934, −5.95555504088419544670041918218, −4.89941034670506776013020192728, −4.03548562764286846850420830588, −3.49221212092090749086588071834, −2.85216751259553436006836719353, −1.72509218443276304869626887101, −1.31418954024752428194182321565,
1.31418954024752428194182321565, 1.72509218443276304869626887101, 2.85216751259553436006836719353, 3.49221212092090749086588071834, 4.03548562764286846850420830588, 4.89941034670506776013020192728, 5.95555504088419544670041918218, 6.67489344114888901496490848934, 7.50307494821165795279248770826, 7.987400731958146934547129163785
