L(s) = 1 | − 7-s − 5·11-s − 13-s − 7·17-s − 6·19-s − 3·23-s + 2·29-s − 2·31-s + 7·37-s − 9·41-s + 8·43-s − 10·47-s − 6·49-s − 5·53-s − 5·61-s + 4·67-s + 9·71-s + 6·73-s + 5·77-s + 3·79-s − 4·83-s − 11·89-s + 91-s + 11·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.50·11-s − 0.277·13-s − 1.69·17-s − 1.37·19-s − 0.625·23-s + 0.371·29-s − 0.359·31-s + 1.15·37-s − 1.40·41-s + 1.21·43-s − 1.45·47-s − 6/7·49-s − 0.686·53-s − 0.640·61-s + 0.488·67-s + 1.06·71-s + 0.702·73-s + 0.569·77-s + 0.337·79-s − 0.439·83-s − 1.16·89-s + 0.104·91-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.18922181147626, −12.93475369420216, −12.62806446614328, −12.01108774250118, −11.34251730880821, −10.90645061356507, −10.68223761732341, −10.04918614914902, −9.605268129308173, −9.160853738909794, −8.425410718412458, −8.168651787596849, −7.762523912044738, −6.939230723710234, −6.627741037652520, −6.165555181158094, −5.555078599774552, −4.935391574963509, −4.505323107890869, −4.068398124942730, −3.231997178745395, −2.735316717972283, −2.141035590994649, −1.786567096206991, −0.5098740736531083, 0,
0.5098740736531083, 1.786567096206991, 2.141035590994649, 2.735316717972283, 3.231997178745395, 4.068398124942730, 4.505323107890869, 4.935391574963509, 5.555078599774552, 6.165555181158094, 6.627741037652520, 6.939230723710234, 7.762523912044738, 8.168651787596849, 8.425410718412458, 9.160853738909794, 9.605268129308173, 10.04918614914902, 10.68223761732341, 10.90645061356507, 11.34251730880821, 12.01108774250118, 12.62806446614328, 12.93475369420216, 13.18922181147626