L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 15-s + 2·17-s + 8·19-s + 4·21-s − 4·23-s + 25-s + 27-s + 6·29-s − 8·31-s − 4·35-s − 6·37-s + 6·41-s + 4·43-s − 45-s + 4·47-s + 9·49-s + 2·51-s + 6·53-s + 8·57-s + 8·59-s − 10·61-s + 4·63-s + 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 1.05·57-s + 1.04·59-s − 1.28·61-s + 0.503·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.583559209\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.583559209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.00634034850278, −13.78158116475816, −13.03140391587681, −12.28611880477285, −12.01437200642815, −11.62676244665418, −10.88612101669202, −10.65814978360349, −9.941514197333107, −9.333047904094248, −8.958127707850947, −8.184521478029343, −8.000472662307790, −7.414412218927116, −7.141189135048139, −6.233334465515831, −5.446896403125793, −5.151689694604651, −4.556397607126668, −3.775485462880403, −3.526326670593522, −2.591963965893027, −2.053146621891313, −1.279506454130694, −0.7370765834934863,
0.7370765834934863, 1.279506454130694, 2.053146621891313, 2.591963965893027, 3.526326670593522, 3.775485462880403, 4.556397607126668, 5.151689694604651, 5.446896403125793, 6.233334465515831, 7.141189135048139, 7.414412218927116, 8.000472662307790, 8.184521478029343, 8.958127707850947, 9.333047904094248, 9.941514197333107, 10.65814978360349, 10.88612101669202, 11.62676244665418, 12.01437200642815, 12.28611880477285, 13.03140391587681, 13.78158116475816, 14.00634034850278