L(s) = 1 | − 9.32·2-s + 9·3-s + 54.9·4-s + 99.7·5-s − 83.9·6-s + 125.·7-s − 214.·8-s + 81·9-s − 930.·10-s + 177.·11-s + 494.·12-s − 919.·13-s − 1.17e3·14-s + 897.·15-s + 240.·16-s + 1.56e3·17-s − 755.·18-s + 447.·19-s + 5.48e3·20-s + 1.13e3·21-s − 1.65e3·22-s − 529·23-s − 1.93e3·24-s + 6.81e3·25-s + 8.57e3·26-s + 729·27-s + 6.92e3·28-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 0.577·3-s + 1.71·4-s + 1.78·5-s − 0.951·6-s + 0.971·7-s − 1.18·8-s + 0.333·9-s − 2.94·10-s + 0.442·11-s + 0.992·12-s − 1.50·13-s − 1.60·14-s + 1.02·15-s + 0.235·16-s + 1.31·17-s − 0.549·18-s + 0.284·19-s + 3.06·20-s + 0.561·21-s − 0.730·22-s − 0.208·23-s − 0.684·24-s + 2.18·25-s + 2.48·26-s + 0.192·27-s + 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.460469041\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460469041\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
good | 2 | \( 1 + 9.32T + 32T^{2} \) |
| 5 | \( 1 - 99.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 177.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 919.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 447.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
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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
−14.16526574554128143465529243572, −12.53128808663571473662143967899, −10.97118806892819696068065747106, −9.746008404494026706243769606036, −9.486359367968845420967693155008, −8.131878632539373894394789052338, −7.04942233489018741394881458200, −5.35898668893587231901003368819, −2.33644919905903927464820475449, −1.37838222334649328233933981507,
1.37838222334649328233933981507, 2.33644919905903927464820475449, 5.35898668893587231901003368819, 7.04942233489018741394881458200, 8.131878632539373894394789052338, 9.486359367968845420967693155008, 9.746008404494026706243769606036, 10.97118806892819696068065747106, 12.53128808663571473662143967899, 14.16526574554128143465529243572