| L(s) = 1 | − 7-s − 3·9-s + 2·11-s − 2·17-s + 23-s − 6·29-s + 10·31-s + 4·37-s − 10·41-s − 8·43-s − 4·47-s + 49-s + 8·53-s − 10·59-s − 2·61-s + 3·63-s + 12·67-s − 8·71-s + 4·73-s − 2·77-s + 14·79-s + 9·81-s − 4·83-s + 10·89-s − 14·97-s − 6·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 0.603·11-s − 0.485·17-s + 0.208·23-s − 1.11·29-s + 1.79·31-s + 0.657·37-s − 1.56·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.09·53-s − 1.30·59-s − 0.256·61-s + 0.377·63-s + 1.46·67-s − 0.949·71-s + 0.468·73-s − 0.227·77-s + 1.57·79-s + 81-s − 0.439·83-s + 1.05·89-s − 1.42·97-s − 0.603·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.47141400311201, −13.95483506421633, −13.40375100795979, −13.20366393107589, −12.34287352714414, −11.93782945615539, −11.44622306028913, −11.13072043984144, −10.30740372784641, −9.991662608893975, −9.212681937978648, −8.982522115972309, −8.254416424474024, −7.953207781770929, −7.112128446867531, −6.489213365676928, −6.295820382595660, −5.509491628016169, −4.969215148686004, −4.364481256024327, −3.565595542745253, −3.179323611674884, −2.434282837689643, −1.773272898041458, −0.8317548326882616, 0,
0.8317548326882616, 1.773272898041458, 2.434282837689643, 3.179323611674884, 3.565595542745253, 4.364481256024327, 4.969215148686004, 5.509491628016169, 6.295820382595660, 6.489213365676928, 7.112128446867531, 7.953207781770929, 8.254416424474024, 8.982522115972309, 9.212681937978648, 9.991662608893975, 10.30740372784641, 11.13072043984144, 11.44622306028913, 11.93782945615539, 12.34287352714414, 13.20366393107589, 13.40375100795979, 13.95483506421633, 14.47141400311201
