| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 19-s + 23-s − 2·31-s − 32-s + 36-s + 37-s − 38-s − 41-s + 43-s − 46-s + 49-s − 53-s + 59-s + 2·62-s + 64-s − 72-s − 73-s − 74-s + 76-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 19-s + 23-s − 2·31-s − 32-s + 36-s + 37-s − 38-s − 41-s + 43-s − 46-s + 49-s − 53-s + 59-s + 2·62-s + 64-s − 72-s − 73-s − 74-s + 76-s + 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9344772231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9344772231\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.879192335081045314695967451042, −7.86977773157743471667263342797, −7.32094667495575994218301220752, −6.82878316619932948871863070073, −5.84327941842390067087537324701, −5.07169282514408694596459572641, −3.92735688759985788013597278188, −3.05662360819855701091263568305, −1.96820460504073601493948205854, −1.02033750853561199502622008593,
1.02033750853561199502622008593, 1.96820460504073601493948205854, 3.05662360819855701091263568305, 3.92735688759985788013597278188, 5.07169282514408694596459572641, 5.84327941842390067087537324701, 6.82878316619932948871863070073, 7.32094667495575994218301220752, 7.86977773157743471667263342797, 8.879192335081045314695967451042
