| L(s) = 1 | + 5.11·2-s + 3.69·3-s + 18.1·4-s + 3.66·5-s + 18.9·6-s + 28.3·7-s + 52.1·8-s − 13.3·9-s + 18.7·10-s + 0.682·11-s + 67.2·12-s + 53.7·13-s + 144.·14-s + 13.5·15-s + 121.·16-s − 109.·17-s − 68.1·18-s − 30.1·19-s + 66.5·20-s + 104.·21-s + 3.49·22-s + 23·23-s + 192.·24-s − 111.·25-s + 274.·26-s − 149.·27-s + 515.·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.711·3-s + 2.27·4-s + 0.327·5-s + 1.28·6-s + 1.52·7-s + 2.30·8-s − 0.493·9-s + 0.592·10-s + 0.0187·11-s + 1.61·12-s + 1.14·13-s + 2.76·14-s + 0.233·15-s + 1.89·16-s − 1.56·17-s − 0.892·18-s − 0.364·19-s + 0.744·20-s + 1.08·21-s + 0.0338·22-s + 0.208·23-s + 1.63·24-s − 0.892·25-s + 2.07·26-s − 1.06·27-s + 3.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.308835696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.308835696\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 5.11T + 8T^{2} \) |
| 3 | \( 1 - 3.69T + 27T^{2} \) |
| 5 | \( 1 - 3.66T + 125T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 - 0.682T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.1T + 6.85e3T^{2} \) |
| 31 | \( 1 + 66.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 498.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 199.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 192.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 101.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 237.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 801.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 210.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 193.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 16.2T + 9.12e5T^{2} \) |
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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
−10.66782704706548726591809248583, −8.966393181807718855147307572498, −8.320411855558312724345678471411, −7.28249814873486063149845217984, −6.19423898763728087859555591536, −5.44233372038378195625802483033, −4.46952308227260143297949186555, −3.72244195744786227114731675372, −2.45710202229460113073999659385, −1.75181071567793654018771642793,
1.75181071567793654018771642793, 2.45710202229460113073999659385, 3.72244195744786227114731675372, 4.46952308227260143297949186555, 5.44233372038378195625802483033, 6.19423898763728087859555591536, 7.28249814873486063149845217984, 8.320411855558312724345678471411, 8.966393181807718855147307572498, 10.66782704706548726591809248583
