| L(s) = 1 | + 3.81·2-s − 8.34·3-s + 6.58·4-s − 2.56·5-s − 31.8·6-s + 3.42·7-s − 5.39·8-s + 42.6·9-s − 9.79·10-s − 24.5·11-s − 54.9·12-s − 46.2·13-s + 13.0·14-s + 21.4·15-s − 73.3·16-s + 97.5·17-s + 163.·18-s + 131.·19-s − 16.8·20-s − 28.5·21-s − 93.6·22-s + 175.·23-s + 45.0·24-s − 118.·25-s − 176.·26-s − 130.·27-s + 22.5·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 1.60·3-s + 0.823·4-s − 0.229·5-s − 2.16·6-s + 0.184·7-s − 0.238·8-s + 1.58·9-s − 0.309·10-s − 0.672·11-s − 1.32·12-s − 0.987·13-s + 0.249·14-s + 0.368·15-s − 1.14·16-s + 1.39·17-s + 2.13·18-s + 1.59·19-s − 0.188·20-s − 0.296·21-s − 0.907·22-s + 1.58·23-s + 0.383·24-s − 0.947·25-s − 1.33·26-s − 0.933·27-s + 0.152·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 3.81T + 8T^{2} \) |
| 3 | \( 1 + 8.34T + 27T^{2} \) |
| 5 | \( 1 + 2.56T + 125T^{2} \) |
| 7 | \( 1 - 3.42T + 343T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 201.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 892.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.66T + 3.00e5T^{2} \) |
| 71 | \( 1 + 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 481.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 296.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 746.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.253159717801776557482735038664, −7.21712038513936969052316254164, −6.73970681291910471233921341133, −5.53981246647270682894707488601, −5.28235654565260329157856797537, −4.81714865040389882435947486198, −3.64367293317627771040631542755, −2.77065503434456314285324853132, −1.13634254768471285528160684803, 0,
1.13634254768471285528160684803, 2.77065503434456314285324853132, 3.64367293317627771040631542755, 4.81714865040389882435947486198, 5.28235654565260329157856797537, 5.53981246647270682894707488601, 6.73970681291910471233921341133, 7.21712038513936969052316254164, 8.253159717801776557482735038664
