| L(s) = 1 | − 2.81·3-s − 3.21·5-s + 2.10·7-s + 4.94·9-s + 6.27·11-s + 2.76·13-s + 9.06·15-s − 2.96·17-s − 6.08·19-s − 5.92·21-s − 3.35·23-s + 5.35·25-s − 5.46·27-s − 7.97·31-s − 17.6·33-s − 6.76·35-s + 4.60·37-s − 7.79·39-s + 1.42·41-s − 3.08·43-s − 15.9·45-s − 0.351·47-s − 2.57·49-s + 8.36·51-s + 6.59·53-s − 20.2·55-s + 17.1·57-s + ⋯ |
| L(s) = 1 | − 1.62·3-s − 1.43·5-s + 0.794·7-s + 1.64·9-s + 1.89·11-s + 0.767·13-s + 2.34·15-s − 0.720·17-s − 1.39·19-s − 1.29·21-s − 0.700·23-s + 1.07·25-s − 1.05·27-s − 1.43·31-s − 3.07·33-s − 1.14·35-s + 0.756·37-s − 1.24·39-s + 0.222·41-s − 0.470·43-s − 2.37·45-s − 0.0513·47-s − 0.368·49-s + 1.17·51-s + 0.905·53-s − 2.72·55-s + 2.26·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 - 2.76T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 0.351T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 9.24T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 3.68T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
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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
−7.45719229605936328884012417271, −6.74588387366449677423218611660, −6.34539225112350067016593573412, −5.56133661303598371141424684928, −4.60149892566180504224923558227, −4.10218994854672689636114762613, −3.74379476975264310922194811761, −1.92044652169018754435997954438, −1.01721949721087926494841467918, 0,
1.01721949721087926494841467918, 1.92044652169018754435997954438, 3.74379476975264310922194811761, 4.10218994854672689636114762613, 4.60149892566180504224923558227, 5.56133661303598371141424684928, 6.34539225112350067016593573412, 6.74588387366449677423218611660, 7.45719229605936328884012417271
