| L(s) = 1 | + 3·7-s − 3·11-s − 13-s − 7·17-s + 8·19-s − 4·23-s − 3·29-s − 11·31-s + 2·41-s + 8·43-s + 9·47-s + 2·49-s + 9·53-s − 9·59-s − 61-s − 5·67-s − 12·73-s − 9·77-s − 8·79-s − 9·83-s − 12·89-s − 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.904·11-s − 0.277·13-s − 1.69·17-s + 1.83·19-s − 0.834·23-s − 0.557·29-s − 1.97·31-s + 0.312·41-s + 1.21·43-s + 1.31·47-s + 2/7·49-s + 1.23·53-s − 1.17·59-s − 0.128·61-s − 0.610·67-s − 1.40·73-s − 1.02·77-s − 0.900·79-s − 0.987·83-s − 1.27·89-s − 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(\approx\) |
\(1.325262260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325262260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.09830993756745, −12.72193499501695, −12.12595825248079, −11.65729608535809, −11.11911645782852, −10.98042794067297, −10.36341913993100, −9.840884930921580, −9.234126033730052, −8.834119655857942, −8.448252883113503, −7.569754016314282, −7.432776769132920, −7.234966812711928, −6.151276955895109, −5.749931718140069, −5.261715746958606, −4.837123451795117, −4.182586079427056, −3.829939378626872, −2.892229406180465, −2.462368012962290, −1.839957576295546, −1.304970870002977, −0.3199692703401142,
0.3199692703401142, 1.304970870002977, 1.839957576295546, 2.462368012962290, 2.892229406180465, 3.829939378626872, 4.182586079427056, 4.837123451795117, 5.261715746958606, 5.749931718140069, 6.151276955895109, 7.234966812711928, 7.432776769132920, 7.569754016314282, 8.448252883113503, 8.834119655857942, 9.234126033730052, 9.840884930921580, 10.36341913993100, 10.98042794067297, 11.11911645782852, 11.65729608535809, 12.12595825248079, 12.72193499501695, 13.09830993756745
