| L(s) = 1 | + 3·3-s − 19·7-s + 9·9-s − 22·11-s − 13-s + 58·17-s + 53·19-s − 57·21-s + 58·23-s + 27·27-s + 22·29-s + 35·31-s − 66·33-s + 270·37-s − 3·39-s − 468·41-s − 431·43-s − 230·47-s + 18·49-s + 174·51-s + 159·57-s − 446·59-s + 127·61-s − 171·63-s − 811·67-s + 174·69-s − 36·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.02·7-s + 1/3·9-s − 0.603·11-s − 0.0213·13-s + 0.827·17-s + 0.639·19-s − 0.592·21-s + 0.525·23-s + 0.192·27-s + 0.140·29-s + 0.202·31-s − 0.348·33-s + 1.19·37-s − 0.0123·39-s − 1.78·41-s − 1.52·43-s − 0.713·47-s + 0.0524·49-s + 0.477·51-s + 0.369·57-s − 0.984·59-s + 0.266·61-s − 0.341·63-s − 1.47·67-s + 0.303·69-s − 0.0601·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 53 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 - 35 T + p^{3} T^{2} \) |
| 37 | \( 1 - 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 468 T + p^{3} T^{2} \) |
| 43 | \( 1 + 431 T + p^{3} T^{2} \) |
| 47 | \( 1 + 230 T + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + 446 T + p^{3} T^{2} \) |
| 61 | \( 1 - 127 T + p^{3} T^{2} \) |
| 67 | \( 1 + 811 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 522 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1368 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1138 T + p^{3} T^{2} \) |
| 89 | \( 1 - 144 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1079 T + p^{3} T^{2} \) |
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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
−8.984632654488731329070125834609, −8.162623227566701788580918216966, −7.36629795609268426179821448955, −6.55694263435804588188905597332, −5.59220549640977603044446756173, −4.61147447469318269333072602266, −3.32324718398762662551997464094, −2.90771848443028003966760237312, −1.44429795336309528455755467534, 0,
1.44429795336309528455755467534, 2.90771848443028003966760237312, 3.32324718398762662551997464094, 4.61147447469318269333072602266, 5.59220549640977603044446756173, 6.55694263435804588188905597332, 7.36629795609268426179821448955, 8.162623227566701788580918216966, 8.984632654488731329070125834609
