| L(s) = 1 | + 10.3·2-s − 9·3-s + 76.0·4-s − 102.·5-s − 93.5·6-s + 106.·7-s + 457.·8-s + 81·9-s − 1.06e3·10-s − 490.·11-s − 684.·12-s + 992.·13-s + 1.10e3·14-s + 924.·15-s + 2.32e3·16-s − 81.9·17-s + 841.·18-s − 13.0·19-s − 7.81e3·20-s − 960.·21-s − 5.09e3·22-s − 4.49e3·23-s − 4.12e3·24-s + 7.43e3·25-s + 1.03e4·26-s − 729·27-s + 8.11e3·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.37·4-s − 1.83·5-s − 1.06·6-s + 0.823·7-s + 2.52·8-s + 0.333·9-s − 3.37·10-s − 1.22·11-s − 1.37·12-s + 1.62·13-s + 1.51·14-s + 1.06·15-s + 2.27·16-s − 0.0687·17-s + 0.612·18-s − 0.00830·19-s − 4.36·20-s − 0.475·21-s − 2.24·22-s − 1.77·23-s − 1.46·24-s + 2.37·25-s + 2.99·26-s − 0.192·27-s + 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 10.3T + 32T^{2} \) |
| 5 | \( 1 + 102.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 490.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 992.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 81.9T + 1.41e6T^{2} \) |
| 19 | \( 1 + 13.0T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.79e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−10.57687438603730530479143386854, −8.281615526461933203997009664710, −7.76866713003604384107964865393, −6.76249564245075735046262526860, −5.66402144498148070768866760099, −4.86983886931989989369144095708, −3.97449834498293291846026255238, −3.39689338540081978307823839648, −1.75710942418682290778309437370, 0,
1.75710942418682290778309437370, 3.39689338540081978307823839648, 3.97449834498293291846026255238, 4.86983886931989989369144095708, 5.66402144498148070768866760099, 6.76249564245075735046262526860, 7.76866713003604384107964865393, 8.281615526461933203997009664710, 10.57687438603730530479143386854
