| L(s) = 1 | − 5.64·3-s − 12.1·5-s − 33.8·7-s + 4.91·9-s + 20.6·13-s + 68.7·15-s + 60.1·17-s + 105.·19-s + 191.·21-s + 186.·23-s + 23.2·25-s + 124.·27-s − 135.·29-s + 89.6·31-s + 412.·35-s − 183.·37-s − 116.·39-s − 372.·41-s − 294.·43-s − 59.8·45-s + 8.47·47-s + 802.·49-s − 339.·51-s − 219.·53-s − 598.·57-s + 260.·59-s + 514.·61-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 1.08·5-s − 1.82·7-s + 0.181·9-s + 0.440·13-s + 1.18·15-s + 0.858·17-s + 1.27·19-s + 1.98·21-s + 1.69·23-s + 0.185·25-s + 0.889·27-s − 0.867·29-s + 0.519·31-s + 1.98·35-s − 0.817·37-s − 0.479·39-s − 1.41·41-s − 1.04·43-s − 0.198·45-s + 0.0263·47-s + 2.33·49-s − 0.933·51-s − 0.569·53-s − 1.39·57-s + 0.574·59-s + 1.08·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 5.64T + 27T^{2} \) |
| 5 | \( 1 + 12.1T + 125T^{2} \) |
| 7 | \( 1 + 33.8T + 343T^{2} \) |
| 13 | \( 1 - 20.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 186.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 89.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 183.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 8.47T + 1.03e5T^{2} \) |
| 53 | \( 1 + 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 260.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 210.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 934.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 238.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 620.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−9.367698632416796455037964875748, −8.374630339502662723126232512963, −7.22098374619536424239544202892, −6.71203552661881784892388461265, −5.75257980302797775298357908967, −5.00273773875573917296600761384, −3.53735200960973284239642405761, −3.18748682264423219084776720327, −0.898603883452229434451006403324, 0,
0.898603883452229434451006403324, 3.18748682264423219084776720327, 3.53735200960973284239642405761, 5.00273773875573917296600761384, 5.75257980302797775298357908967, 6.71203552661881784892388461265, 7.22098374619536424239544202892, 8.374630339502662723126232512963, 9.367698632416796455037964875748
