| L(s) = 1 | − 4·2-s + 27.8·3-s + 16·4-s − 69.2·5-s − 111.·6-s − 195.·7-s − 64·8-s + 531.·9-s + 276.·10-s − 137.·11-s + 445.·12-s + 796.·13-s + 781.·14-s − 1.92e3·15-s + 256·16-s + 1.56e3·17-s − 2.12e3·18-s − 1.10e3·20-s − 5.43e3·21-s + 549.·22-s − 417.·23-s − 1.78e3·24-s + 1.66e3·25-s − 3.18e3·26-s + 8.04e3·27-s − 3.12e3·28-s − 311.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·3-s + 0.5·4-s − 1.23·5-s − 1.26·6-s − 1.50·7-s − 0.353·8-s + 2.18·9-s + 0.875·10-s − 0.342·11-s + 0.892·12-s + 1.30·13-s + 1.06·14-s − 2.21·15-s + 0.250·16-s + 1.31·17-s − 1.54·18-s − 0.619·20-s − 2.69·21-s + 0.242·22-s − 0.164·23-s − 0.631·24-s + 0.532·25-s − 0.924·26-s + 2.12·27-s − 0.753·28-s − 0.0688·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 27.8T + 243T^{2} \) |
| 5 | \( 1 + 69.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 137.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 796.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.56e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 417.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 311.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.17e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.03e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.72e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.62e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.30e5T + 8.58e9T^{2} \) |
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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
−9.028188875302365241803275874076, −8.464866706933413906587940007056, −7.61315034939908815390742865598, −7.17071431312063845565562509544, −5.91172979482718588468873210662, −3.88986837979218303067894877071, −3.51383911014598339305554471961, −2.74017769791885801973916173717, −1.31379567027024246227076424350, 0,
1.31379567027024246227076424350, 2.74017769791885801973916173717, 3.51383911014598339305554471961, 3.88986837979218303067894877071, 5.91172979482718588468873210662, 7.17071431312063845565562509544, 7.61315034939908815390742865598, 8.464866706933413906587940007056, 9.028188875302365241803275874076
