| L(s) = 1 | + 22·3-s − 25·5-s + 218·7-s + 241·9-s − 480·11-s − 622·13-s − 550·15-s + 186·17-s − 1.20e3·19-s + 4.79e3·21-s − 3.18e3·23-s + 625·25-s − 44·27-s + 5.52e3·29-s + 9.35e3·31-s − 1.05e4·33-s − 5.45e3·35-s + 5.61e3·37-s − 1.36e4·39-s − 1.43e4·41-s − 370·43-s − 6.02e3·45-s + 1.61e4·47-s + 3.07e4·49-s + 4.09e3·51-s − 4.37e3·53-s + 1.20e4·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 0.447·5-s + 1.68·7-s + 0.991·9-s − 1.19·11-s − 1.02·13-s − 0.631·15-s + 0.156·17-s − 0.765·19-s + 2.37·21-s − 1.25·23-s + 1/5·25-s − 0.0116·27-s + 1.22·29-s + 1.74·31-s − 1.68·33-s − 0.752·35-s + 0.674·37-s − 1.44·39-s − 1.33·41-s − 0.0305·43-s − 0.443·45-s + 1.06·47-s + 1.82·49-s + 0.220·51-s − 0.213·53-s + 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.996582257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996582257\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 22 T + p^{5} T^{2} \) |
| 7 | \( 1 - 218 T + p^{5} T^{2} \) |
| 11 | \( 1 + 480 T + p^{5} T^{2} \) |
| 13 | \( 1 + 622 T + p^{5} T^{2} \) |
| 17 | \( 1 - 186 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1204 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3186 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9356 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5618 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14394 T + p^{5} T^{2} \) |
| 43 | \( 1 + 370 T + p^{5} T^{2} \) |
| 47 | \( 1 - 16146 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 + 11748 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13202 T + p^{5} T^{2} \) |
| 67 | \( 1 + 11542 T + p^{5} T^{2} \) |
| 71 | \( 1 + 29532 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33698 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31208 T + p^{5} T^{2} \) |
| 83 | \( 1 + 38466 T + p^{5} T^{2} \) |
| 89 | \( 1 - 119514 T + p^{5} T^{2} \) |
| 97 | \( 1 - 94658 T + p^{5} T^{2} \) |
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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
−17.45760592998863266538651229557, −15.53456684155579359953972139624, −14.66844824519942955845592312212, −13.72210593724656417482917285761, −11.99534170829726721147213851992, −10.29024041268297841738990171808, −8.362876039403005985060153086809, −7.79164436867101309506219019452, −4.63666620757170252210011277593, −2.39577658256314220724904231661,
2.39577658256314220724904231661, 4.63666620757170252210011277593, 7.79164436867101309506219019452, 8.362876039403005985060153086809, 10.29024041268297841738990171808, 11.99534170829726721147213851992, 13.72210593724656417482917285761, 14.66844824519942955845592312212, 15.53456684155579359953972139624, 17.45760592998863266538651229557
