| L(s) = 1 | + 2.56·3-s + 3.56·9-s − 2.56·11-s − 5.68·13-s + 3.43·17-s − 1.12·19-s + 5.12·23-s + 1.43·27-s + 4.56·29-s + 10.2·31-s − 6.56·33-s − 8.24·37-s − 14.5·39-s − 7.12·41-s − 1.12·43-s + 6.56·47-s + 8.80·51-s + 4.87·53-s − 2.87·57-s + 4·59-s + 15.1·61-s + 14.2·67-s + 13.1·69-s + 12.2·73-s − 11.6·79-s − 6.99·81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.18·9-s − 0.772·11-s − 1.57·13-s + 0.833·17-s − 0.257·19-s + 1.06·23-s + 0.276·27-s + 0.847·29-s + 1.84·31-s − 1.14·33-s − 1.35·37-s − 2.33·39-s − 1.11·41-s − 0.171·43-s + 0.957·47-s + 1.23·51-s + 0.669·53-s − 0.381·57-s + 0.520·59-s + 1.93·61-s + 1.74·67-s + 1.57·69-s + 1.43·73-s − 1.31·79-s − 0.777·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364099778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364099778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.86217499834449785748166201244, −7.09649512697186944939616124452, −6.65418378914650171822743020873, −5.29175642791039600895899975568, −5.01092922542570892458922482269, −4.02038187091920110127114256360, −3.20649409371169983798220706738, −2.62557651961971797966650113412, −2.10032276905432378676392079720, −0.790127328703066710935470922177,
0.790127328703066710935470922177, 2.10032276905432378676392079720, 2.62557651961971797966650113412, 3.20649409371169983798220706738, 4.02038187091920110127114256360, 5.01092922542570892458922482269, 5.29175642791039600895899975568, 6.65418378914650171822743020873, 7.09649512697186944939616124452, 7.86217499834449785748166201244
