| L(s) = 1 | + 2-s + 3-s + 4-s + 3.71·5-s + 6-s − 4.60·7-s + 8-s + 9-s + 3.71·10-s + 2.78·11-s + 12-s + 1.37·13-s − 4.60·14-s + 3.71·15-s + 16-s + 4.93·17-s + 18-s − 7.94·19-s + 3.71·20-s − 4.60·21-s + 2.78·22-s + 23-s + 24-s + 8.82·25-s + 1.37·26-s + 27-s − 4.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.66·5-s + 0.408·6-s − 1.74·7-s + 0.353·8-s + 0.333·9-s + 1.17·10-s + 0.838·11-s + 0.288·12-s + 0.382·13-s − 1.23·14-s + 0.960·15-s + 0.250·16-s + 1.19·17-s + 0.235·18-s − 1.82·19-s + 0.831·20-s − 1.00·21-s + 0.592·22-s + 0.208·23-s + 0.204·24-s + 1.76·25-s + 0.270·26-s + 0.192·27-s − 0.870·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.750390754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.750390754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 7.94T + 19T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 + 0.204T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 + 9.98T + 97T^{2} \) |
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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
−8.675721573297086754333989159981, −7.52786583352739567846580266586, −6.59757648005080965320546947189, −6.12545088621105549713255873797, −5.87204758830449268438512900988, −4.56444185656172689283646623128, −3.73459347337016259018200106871, −2.91928288832540979581980285398, −2.28680206260758040551851409909, −1.17660033937910510025208369478,
1.17660033937910510025208369478, 2.28680206260758040551851409909, 2.91928288832540979581980285398, 3.73459347337016259018200106871, 4.56444185656172689283646623128, 5.87204758830449268438512900988, 6.12545088621105549713255873797, 6.59757648005080965320546947189, 7.52786583352739567846580266586, 8.675721573297086754333989159981
