| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 13-s + 15-s + 2·17-s − 4·19-s − 4·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 4·35-s + 10·37-s + 39-s + 10·41-s − 4·43-s − 45-s + 9·49-s − 2·51-s + 10·53-s + 4·57-s − 12·59-s + 2·61-s + 4·63-s + 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.676·35-s + 1.64·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.280·51-s + 1.37·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.21389891319681, −12.86238570257180, −12.22774274493645, −11.94008994404102, −11.44058314477510, −11.05471622895688, −10.67705056547283, −10.25290255812056, −9.412838958636345, −9.263810738126254, −8.362053018126838, −8.028400404670144, −7.704523468624123, −7.226248158667255, −6.533212565592400, −6.004205094984840, −5.495175227330318, −5.049416965106498, −4.374774712083545, −4.173681396984093, −3.552955941452582, −2.559885404663863, −2.143659607594128, −1.439659823769605, −0.8300720554274437, 0,
0.8300720554274437, 1.439659823769605, 2.143659607594128, 2.559885404663863, 3.552955941452582, 4.173681396984093, 4.374774712083545, 5.049416965106498, 5.495175227330318, 6.004205094984840, 6.533212565592400, 7.226248158667255, 7.704523468624123, 8.028400404670144, 8.362053018126838, 9.263810738126254, 9.412838958636345, 10.25290255812056, 10.67705056547283, 11.05471622895688, 11.44058314477510, 11.94008994404102, 12.22774274493645, 12.86238570257180, 13.21389891319681
