| L(s) = 1 | − 2·4-s + 3·7-s + 4·11-s − 13-s + 4·16-s + 4·17-s − 5·19-s − 4·23-s − 6·28-s − 8·29-s + 5·31-s − 37-s − 8·41-s − 9·43-s − 8·44-s − 12·47-s + 2·49-s + 2·52-s + 4·59-s − 61-s − 8·64-s − 5·67-s − 8·68-s − 4·71-s − 6·73-s + 10·76-s + 12·77-s + ⋯ |
| L(s) = 1 | − 4-s + 1.13·7-s + 1.20·11-s − 0.277·13-s + 16-s + 0.970·17-s − 1.14·19-s − 0.834·23-s − 1.13·28-s − 1.48·29-s + 0.898·31-s − 0.164·37-s − 1.24·41-s − 1.37·43-s − 1.20·44-s − 1.75·47-s + 2/7·49-s + 0.277·52-s + 0.520·59-s − 0.128·61-s − 64-s − 0.610·67-s − 0.970·68-s − 0.474·71-s − 0.702·73-s + 1.14·76-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.73399753957985416155007133298, −6.70846979961600727854791898641, −6.04428051711941524391541531809, −5.15757924620524458245492034731, −4.71285875378596168196506835357, −3.92778574723351530218672285521, −3.38077332567254002616127775485, −1.93349418455518354421482928110, −1.33130240711472725214006926940, 0,
1.33130240711472725214006926940, 1.93349418455518354421482928110, 3.38077332567254002616127775485, 3.92778574723351530218672285521, 4.71285875378596168196506835357, 5.15757924620524458245492034731, 6.04428051711941524391541531809, 6.70846979961600727854791898641, 7.73399753957985416155007133298
