| L(s) = 1 | + 3·5-s + 17-s − 4·23-s + 4·25-s + 3·29-s + 8·31-s + 5·37-s − 3·41-s − 4·43-s + 8·47-s − 7·49-s − 13·53-s + 12·59-s − 15·61-s − 12·67-s − 8·71-s + 3·73-s − 4·79-s + 12·83-s + 3·85-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.242·17-s − 0.834·23-s + 4/5·25-s + 0.557·29-s + 1.43·31-s + 0.821·37-s − 0.468·41-s − 0.609·43-s + 1.16·47-s − 49-s − 1.78·53-s + 1.56·59-s − 1.92·61-s − 1.46·67-s − 0.949·71-s + 0.351·73-s − 0.450·79-s + 1.31·83-s + 0.325·85-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.256146327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.256146327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.67386810724628, −13.45610031405698, −12.89755438176545, −12.33552413120533, −11.85081464342935, −11.40206278008542, −10.63233688310899, −10.25677731741939, −9.878257899636397, −9.413705410543603, −8.901817997869800, −8.280689290470427, −7.832923890057737, −7.208297664883781, −6.420250232006226, −6.194776690936109, −5.760758146849839, −4.984910110270338, −4.625499208593300, −3.914787723784986, −3.025842527170624, −2.701390117975779, −1.856105699466378, −1.459243476848939, −0.5581994724009074,
0.5581994724009074, 1.459243476848939, 1.856105699466378, 2.701390117975779, 3.025842527170624, 3.914787723784986, 4.625499208593300, 4.984910110270338, 5.760758146849839, 6.194776690936109, 6.420250232006226, 7.208297664883781, 7.832923890057737, 8.280689290470427, 8.901817997869800, 9.413705410543603, 9.878257899636397, 10.25677731741939, 10.63233688310899, 11.40206278008542, 11.85081464342935, 12.33552413120533, 12.89755438176545, 13.45610031405698, 13.67386810724628
