| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 4·13-s − 4·19-s − 8·21-s − 6·23-s + 4·27-s + 6·29-s − 8·31-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s + 6·53-s + 8·57-s + 12·59-s − 2·61-s + 4·63-s − 10·67-s + 12·69-s + 12·71-s − 16·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s − 1.74·21-s − 1.25·23-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.503·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.87·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.77460400695624, −14.38847750229865, −13.91064290203226, −13.23009482774338, −12.53137105328466, −12.05383601733192, −11.76894369271961, −11.32394863941897, −10.68270478366850, −10.31039649382085, −9.900079738825477, −8.873577404184344, −8.529204932734268, −7.966836574170328, −7.305996627941133, −6.857982526703165, −6.135256103992602, −5.571158209593372, −5.097780455156352, −4.623900861143225, −4.137612519466467, −3.201235450709064, −2.123024096748115, −1.890613716185070, −0.8054225407273239, 0,
0.8054225407273239, 1.890613716185070, 2.123024096748115, 3.201235450709064, 4.137612519466467, 4.623900861143225, 5.097780455156352, 5.571158209593372, 6.135256103992602, 6.857982526703165, 7.305996627941133, 7.966836574170328, 8.529204932734268, 8.873577404184344, 9.900079738825477, 10.31039649382085, 10.68270478366850, 11.32394863941897, 11.76894369271961, 12.05383601733192, 12.53137105328466, 13.23009482774338, 13.91064290203226, 14.38847750229865, 14.77460400695624
