L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s − 4·11-s + 2·15-s − 6·17-s − 19-s − 2·21-s − 23-s − 4·25-s + 4·27-s − 3·29-s − 7·31-s + 8·33-s − 35-s − 10·37-s + 10·41-s − 7·43-s − 45-s − 9·47-s + 49-s + 12·51-s − 3·53-s + 4·55-s + 2·57-s − 6·61-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.516·15-s − 1.45·17-s − 0.229·19-s − 0.436·21-s − 0.208·23-s − 4/5·25-s + 0.769·27-s − 0.557·29-s − 1.25·31-s + 1.39·33-s − 0.169·35-s − 1.64·37-s + 1.56·41-s − 1.06·43-s − 0.149·45-s − 1.31·47-s + 1/7·49-s + 1.68·51-s − 0.412·53-s + 0.539·55-s + 0.264·57-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.25625301349306, −13.77672075031720, −13.12314252884360, −12.78323692526150, −12.27519762451560, −11.71173963524459, −11.20580043272787, −10.87979388182709, −10.64708630986404, −9.836392155762706, −9.336077493284768, −8.596123233910473, −8.165668919777349, −7.674422640829624, −6.953196358027110, −6.643918810833587, −5.822860421602692, −5.497721159338887, −4.895351148730369, −4.490109574493096, −3.751506125837047, −3.099193539583866, −2.153471725683330, −1.780676784005446, −0.5117044006630216, 0,
0.5117044006630216, 1.780676784005446, 2.153471725683330, 3.099193539583866, 3.751506125837047, 4.490109574493096, 4.895351148730369, 5.497721159338887, 5.822860421602692, 6.643918810833587, 6.953196358027110, 7.674422640829624, 8.165668919777349, 8.596123233910473, 9.336077493284768, 9.836392155762706, 10.64708630986404, 10.87979388182709, 11.20580043272787, 11.71173963524459, 12.27519762451560, 12.78323692526150, 13.12314252884360, 13.77672075031720, 14.25625301349306