| L(s) = 1 | + 0.724·2-s + 1.92·3-s − 1.47·4-s − 1.17·5-s + 1.39·6-s − 4.09·7-s − 2.51·8-s + 0.709·9-s − 0.853·10-s − 3.08·11-s − 2.84·12-s + 2.42·13-s − 2.96·14-s − 2.27·15-s + 1.12·16-s + 0.237·17-s + 0.513·18-s − 2.02·19-s + 1.73·20-s − 7.87·21-s − 2.23·22-s − 23-s − 4.84·24-s − 3.61·25-s + 1.75·26-s − 4.41·27-s + 6.03·28-s + ⋯ |
| L(s) = 1 | + 0.512·2-s + 1.11·3-s − 0.737·4-s − 0.527·5-s + 0.569·6-s − 1.54·7-s − 0.889·8-s + 0.236·9-s − 0.269·10-s − 0.930·11-s − 0.820·12-s + 0.673·13-s − 0.791·14-s − 0.586·15-s + 0.282·16-s + 0.0575·17-s + 0.121·18-s − 0.464·19-s + 0.388·20-s − 1.71·21-s − 0.476·22-s − 0.208·23-s − 0.989·24-s − 0.722·25-s + 0.344·26-s − 0.849·27-s + 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.724T + 2T^{2} \) |
| 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 - 0.237T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 31 | \( 1 - 0.00576T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 0.167T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 2.09T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−9.799527862874899643432532956472, −9.207342211992751111748447623687, −8.397524327082498545398646899503, −7.69104718164239574444891994006, −6.39336971944166937058514168436, −5.51310894825833864428775027962, −4.09035269143656761080859218876, −3.46680428167158991295790246357, −2.64227707730943926991536161550, 0,
2.64227707730943926991536161550, 3.46680428167158991295790246357, 4.09035269143656761080859218876, 5.51310894825833864428775027962, 6.39336971944166937058514168436, 7.69104718164239574444891994006, 8.397524327082498545398646899503, 9.207342211992751111748447623687, 9.799527862874899643432532956472
