| L(s) = 1 | + 14.1·2-s − 81·3-s − 313.·4-s − 1.99e3·5-s − 1.14e3·6-s + 5.78e3·7-s − 1.16e4·8-s + 6.56e3·9-s − 2.80e4·10-s − 5.53e4·11-s + 2.53e4·12-s − 1.79e5·13-s + 8.16e4·14-s + 1.61e5·15-s − 3.74e3·16-s + 4.08e5·17-s + 9.25e4·18-s + 1.30e5·19-s + 6.23e5·20-s − 4.68e5·21-s − 7.81e5·22-s + 2.42e6·23-s + 9.42e5·24-s + 2.01e6·25-s − 2.52e6·26-s − 5.31e5·27-s − 1.81e6·28-s + ⋯ |
| L(s) = 1 | + 0.623·2-s − 0.577·3-s − 0.611·4-s − 1.42·5-s − 0.359·6-s + 0.910·7-s − 1.00·8-s + 0.333·9-s − 0.887·10-s − 1.14·11-s + 0.353·12-s − 1.73·13-s + 0.567·14-s + 0.822·15-s − 0.0142·16-s + 1.18·17-s + 0.207·18-s + 0.229·19-s + 0.871·20-s − 0.525·21-s − 0.710·22-s + 1.80·23-s + 0.579·24-s + 1.02·25-s − 1.08·26-s − 0.192·27-s − 0.557·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9160974362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9160974362\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 - 14.1T + 512T^{2} \) |
| 5 | \( 1 + 1.99e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.78e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.53e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.79e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.08e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.26e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.00e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.11e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.14e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.65e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.04e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.39e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.64e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.88e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.97e8T + 7.60e17T^{2} \) |
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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
−13.05797600200556136503749635467, −12.15137625007035227542932250628, −11.38450510432387763910714125167, −9.948305263429299756979393855776, −8.217038594124342982049119801465, −7.35434734979526962443058828228, −5.16151561385744866670799514639, −4.71403577955576008349055984938, −3.11651361435607347901695697439, −0.55756096172790935185096040134,
0.55756096172790935185096040134, 3.11651361435607347901695697439, 4.71403577955576008349055984938, 5.16151561385744866670799514639, 7.35434734979526962443058828228, 8.217038594124342982049119801465, 9.948305263429299756979393855776, 11.38450510432387763910714125167, 12.15137625007035227542932250628, 13.05797600200556136503749635467
