| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s − 11-s − 15-s + 5·19-s − 3·21-s − 4·23-s + 25-s + 27-s + 10·31-s − 33-s + 3·35-s + 37-s + 6·41-s + 2·43-s − 45-s − 9·47-s + 2·49-s + 13·53-s + 55-s + 5·57-s − 4·59-s + 2·61-s − 3·63-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 1.14·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.79·31-s − 0.174·33-s + 0.507·35-s + 0.164·37-s + 0.937·41-s + 0.304·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s + 1.78·53-s + 0.134·55-s + 0.662·57-s − 0.520·59-s + 0.256·61-s − 0.377·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224612567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224612567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.13160662690241, −13.02749919958187, −12.22596604238082, −11.99274232021638, −11.43010000871602, −10.91588223312127, −10.06518366045665, −9.955386263384124, −9.631748448265128, −8.795488100574140, −8.559572040058994, −7.889054163161509, −7.515656192763264, −6.975433953709011, −6.497669092978890, −5.879885681391656, −5.478735898070949, −4.565719134327023, −4.289064055599398, −3.548704472146039, −3.078188273422971, −2.721394266705273, −1.990309788894255, −1.085418671280171, −0.4644045503821014,
0.4644045503821014, 1.085418671280171, 1.990309788894255, 2.721394266705273, 3.078188273422971, 3.548704472146039, 4.289064055599398, 4.565719134327023, 5.478735898070949, 5.879885681391656, 6.497669092978890, 6.975433953709011, 7.515656192763264, 7.889054163161509, 8.559572040058994, 8.795488100574140, 9.631748448265128, 9.955386263384124, 10.06518366045665, 10.91588223312127, 11.43010000871602, 11.99274232021638, 12.22596604238082, 13.02749919958187, 13.13160662690241
