| L(s) = 1 | − 4.80·2-s − 5.52·3-s + 15.0·4-s + 18.8·5-s + 26.5·6-s + 0.563·7-s − 33.8·8-s + 3.56·9-s − 90.3·10-s − 21.9·11-s − 83.2·12-s + 90.1·13-s − 2.70·14-s − 103.·15-s + 42.1·16-s − 68.5·17-s − 17.1·18-s + 29.2·19-s + 283.·20-s − 3.11·21-s + 105.·22-s − 171.·23-s + 187.·24-s + 228.·25-s − 432.·26-s + 129.·27-s + 8.48·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.06·3-s + 1.88·4-s + 1.68·5-s + 1.80·6-s + 0.0304·7-s − 1.49·8-s + 0.132·9-s − 2.85·10-s − 0.601·11-s − 2.00·12-s + 1.92·13-s − 0.0516·14-s − 1.79·15-s + 0.658·16-s − 0.978·17-s − 0.224·18-s + 0.352·19-s + 3.16·20-s − 0.0323·21-s + 1.02·22-s − 1.55·23-s + 1.59·24-s + 1.83·25-s − 3.26·26-s + 0.923·27-s + 0.0572·28-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 | \( 1 + 4.80T + 8T^{2} \) |
| 3 | \( 1 + 5.52T + 27T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 - 0.563T + 343T^{2} \) |
| 11 | \( 1 + 21.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.36T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 191.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 341.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 319.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 51.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 733.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 417.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 35.6T + 3.89e5T^{2} \) |
| 79 | \( 1 - 32.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 73.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 309.T + 9.12e5T^{2} \) |
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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
−8.558959161369235616485322322983, −8.055640391653353082869864794004, −6.66499154758209743975553419439, −6.24249856027295632024338323149, −5.79376935985151122461053136627, −4.65864754523927096471986533077, −2.87837255021818560163952190096, −1.83679200927478058004579943903, −1.13547548158704060747843006525, 0,
1.13547548158704060747843006525, 1.83679200927478058004579943903, 2.87837255021818560163952190096, 4.65864754523927096471986533077, 5.79376935985151122461053136627, 6.24249856027295632024338323149, 6.66499154758209743975553419439, 8.055640391653353082869864794004, 8.558959161369235616485322322983
