| L(s) = 1 | + 5-s − 6·11-s + 4·13-s − 17-s − 19-s − 3·23-s − 4·25-s + 6·29-s − 3·37-s + 12·41-s − 3·43-s − 10·47-s + 2·53-s − 6·55-s + 13·59-s − 10·61-s + 4·65-s + 3·67-s − 3·71-s − 4·73-s − 16·79-s − 12·83-s − 85-s − 89-s − 95-s + 16·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.80·11-s + 1.10·13-s − 0.242·17-s − 0.229·19-s − 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.493·37-s + 1.87·41-s − 0.457·43-s − 1.45·47-s + 0.274·53-s − 0.809·55-s + 1.69·59-s − 1.28·61-s + 0.496·65-s + 0.366·67-s − 0.356·71-s − 0.468·73-s − 1.80·79-s − 1.31·83-s − 0.108·85-s − 0.105·89-s − 0.102·95-s + 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.69301322506156, −13.28199996325081, −12.97041002276723, −12.54079936084577, −11.76523699846468, −11.41308901976401, −10.77294958047411, −10.42669926306392, −10.00398687436072, −9.525496868666685, −8.817513452193382, −8.189169102591125, −8.173361850841840, −7.370009802092457, −6.855482596539036, −6.158428270030403, −5.732367186918244, −5.426579032934738, −4.518781610438055, −4.300309251140437, −3.327832265327461, −2.928803565684923, −2.204390174078659, −1.735750176686110, −0.8063210321544611, 0,
0.8063210321544611, 1.735750176686110, 2.204390174078659, 2.928803565684923, 3.327832265327461, 4.300309251140437, 4.518781610438055, 5.426579032934738, 5.732367186918244, 6.158428270030403, 6.855482596539036, 7.370009802092457, 8.173361850841840, 8.189169102591125, 8.817513452193382, 9.525496868666685, 10.00398687436072, 10.42669926306392, 10.77294958047411, 11.41308901976401, 11.76523699846468, 12.54079936084577, 12.97041002276723, 13.28199996325081, 13.69301322506156
