| L(s) = 1 | − 2·3-s + 6·7-s − 23·9-s + 32·11-s − 38·13-s − 26·17-s + 100·19-s − 12·21-s − 78·23-s + 100·27-s + 50·29-s + 108·31-s − 64·33-s + 266·37-s + 76·39-s + 22·41-s − 442·43-s − 514·47-s − 307·49-s + 52·51-s + 2·53-s − 200·57-s + 500·59-s + 518·61-s − 138·63-s − 126·67-s + 156·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.323·7-s − 0.851·9-s + 0.877·11-s − 0.810·13-s − 0.370·17-s + 1.20·19-s − 0.124·21-s − 0.707·23-s + 0.712·27-s + 0.320·29-s + 0.625·31-s − 0.337·33-s + 1.18·37-s + 0.312·39-s + 0.0838·41-s − 1.56·43-s − 1.59·47-s − 0.895·49-s + 0.142·51-s + 0.00518·53-s − 0.464·57-s + 1.10·59-s + 1.08·61-s − 0.275·63-s − 0.229·67-s + 0.272·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 514 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 500 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 126 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 878 T + p^{3} T^{2} \) |
| 79 | \( 1 + 600 T + p^{3} T^{2} \) |
| 83 | \( 1 + 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
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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
−8.549635747050430196113879821730, −7.959014958217588211203349273307, −6.91251799696926957634376922130, −6.24093483968302457641386762894, −5.30324480511294971580678990386, −4.61968981095360200289027508977, −3.47865249188224677109850834953, −2.47432493945510607249898334872, −1.22786431127161040378292335997, 0,
1.22786431127161040378292335997, 2.47432493945510607249898334872, 3.47865249188224677109850834953, 4.61968981095360200289027508977, 5.30324480511294971580678990386, 6.24093483968302457641386762894, 6.91251799696926957634376922130, 7.959014958217588211203349273307, 8.549635747050430196113879821730
