| L(s) = 1 | + 0.779·2-s + 32.1·3-s − 511.·4-s − 894.·5-s + 25.0·6-s + 2.72e3·7-s − 797.·8-s − 1.86e4·9-s − 697.·10-s + 1.86e4·11-s − 1.64e4·12-s + 5.03e4·13-s + 2.12e3·14-s − 2.88e4·15-s + 2.61e5·16-s − 1.45e4·18-s − 1.06e5·19-s + 4.57e5·20-s + 8.77e4·21-s + 1.45e4·22-s − 1.33e6·23-s − 2.56e4·24-s − 1.15e6·25-s + 3.92e4·26-s − 1.23e6·27-s − 1.39e6·28-s − 3.99e6·29-s + ⋯ |
| L(s) = 1 | + 0.0344·2-s + 0.229·3-s − 0.998·4-s − 0.640·5-s + 0.00790·6-s + 0.429·7-s − 0.0688·8-s − 0.947·9-s − 0.0220·10-s + 0.383·11-s − 0.229·12-s + 0.488·13-s + 0.0147·14-s − 0.146·15-s + 0.996·16-s − 0.0326·18-s − 0.188·19-s + 0.639·20-s + 0.0984·21-s + 0.0132·22-s − 0.992·23-s − 0.0158·24-s − 0.590·25-s + 0.0168·26-s − 0.446·27-s − 0.428·28-s − 1.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8374408064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8374408064\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 0.779T + 512T^{2} \) |
| 3 | \( 1 - 32.1T + 1.96e4T^{2} \) |
| 5 | \( 1 + 894.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.72e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.86e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.03e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.06e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.33e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.47e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.90e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.29e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.35e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.95e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.21e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.11e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.77e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.69e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.17e8T + 7.60e17T^{2} \) |
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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
−10.11084671154755075071273840286, −9.093958336197969465352111688923, −8.353065554383612941623605726214, −7.71896544266099722766425444134, −6.17406390249283673135674503003, −5.19528090317503434204542278307, −4.06997632297626705420722992513, −3.37595999684300852230176238079, −1.80089348576340131622034255675, −0.39899556401412312259355188628,
0.39899556401412312259355188628, 1.80089348576340131622034255675, 3.37595999684300852230176238079, 4.06997632297626705420722992513, 5.19528090317503434204542278307, 6.17406390249283673135674503003, 7.71896544266099722766425444134, 8.353065554383612941623605726214, 9.093958336197969465352111688923, 10.11084671154755075071273840286
