L(s) = 1 | − 3-s − 7-s + 9-s + 6·11-s + 2·13-s + 4·19-s + 21-s + 6·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 6·33-s + 2·37-s − 2·39-s + 12·41-s + 4·43-s − 12·47-s + 49-s − 6·53-s − 4·57-s − 10·61-s − 63-s − 8·67-s − 6·69-s − 6·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.917·19-s + 0.218·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 1.04·33-s + 0.328·37-s − 0.320·39-s + 1.87·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.977·67-s − 0.722·69-s − 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.194939088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194939088\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.54560708712528768168345903978, −10.82794086663248344608690404212, −9.542934180818189537522408595040, −9.052436712736537682628040663330, −7.58881932943940700511157248554, −6.58911559432542935222512700270, −5.84091064569873502208467244139, −4.46733872516772697970621091271, −3.36010318341016445358722759890, −1.28052617279627540057290389597,
1.28052617279627540057290389597, 3.36010318341016445358722759890, 4.46733872516772697970621091271, 5.84091064569873502208467244139, 6.58911559432542935222512700270, 7.58881932943940700511157248554, 9.052436712736537682628040663330, 9.542934180818189537522408595040, 10.82794086663248344608690404212, 11.54560708712528768168345903978