L(s) = 1 | − 5.54·2-s + 22.7·4-s − 25.7·7-s − 81.7·8-s − 6.25·11-s + 15.2·13-s + 142.·14-s + 271.·16-s − 36.0·17-s − 52.7·19-s + 34.6·22-s − 83.7·23-s − 84.6·26-s − 584.·28-s − 119.·29-s + 276.·31-s − 849.·32-s + 199.·34-s + 117.·37-s + 292.·38-s + 159.·41-s + 294.·43-s − 142.·44-s + 464.·46-s + 83.2·47-s + 318.·49-s + 346.·52-s + ⋯ |
L(s) = 1 | − 1.96·2-s + 2.84·4-s − 1.38·7-s − 3.61·8-s − 0.171·11-s + 0.325·13-s + 2.72·14-s + 4.23·16-s − 0.513·17-s − 0.636·19-s + 0.336·22-s − 0.759·23-s − 0.638·26-s − 3.94·28-s − 0.762·29-s + 1.60·31-s − 4.69·32-s + 1.00·34-s + 0.522·37-s + 1.24·38-s + 0.606·41-s + 1.04·43-s − 0.487·44-s + 1.48·46-s + 0.258·47-s + 0.927·49-s + 0.925·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 5.54T + 8T^{2} \) |
| 7 | \( 1 + 25.7T + 343T^{2} \) |
| 11 | \( 1 + 6.25T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 83.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 635.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 597.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 330.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 130.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 369.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 225.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 11.9T + 9.12e5T^{2} \) |
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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
−8.467140050375508582488172547343, −7.86933339589324230169405878093, −6.93688317749217786667623981126, −6.38966679242229755798199360988, −5.78287294147640288613307947410, −3.97042652740245127562257509866, −2.87629457116545070891221000050, −2.17704435170297860396787945654, −0.864771407927055956235531452416, 0,
0.864771407927055956235531452416, 2.17704435170297860396787945654, 2.87629457116545070891221000050, 3.97042652740245127562257509866, 5.78287294147640288613307947410, 6.38966679242229755798199360988, 6.93688317749217786667623981126, 7.86933339589324230169405878093, 8.467140050375508582488172547343