| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 4·11-s − 6·13-s + 2·15-s + 2·17-s + 19-s + 21-s + 4·23-s − 25-s + 27-s + 6·29-s − 4·31-s + 4·33-s + 2·35-s + 2·37-s − 6·39-s + 10·41-s − 12·43-s + 2·45-s − 4·47-s + 49-s + 2·51-s + 14·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.338·35-s + 0.328·37-s − 0.960·39-s + 1.56·41-s − 1.82·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 1.92·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.455946024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.455946024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.050882121400679747355536858476, −7.22871495038663873112013199388, −6.80274783066483389766425361700, −5.86954646759091444473031262847, −5.13286708747725914438118767083, −4.47077941170628328261013401345, −3.54125722323614090063054391624, −2.61946596669972812099801775138, −1.94435285936221864504020005068, −0.985326867937882310957049618848,
0.985326867937882310957049618848, 1.94435285936221864504020005068, 2.61946596669972812099801775138, 3.54125722323614090063054391624, 4.47077941170628328261013401345, 5.13286708747725914438118767083, 5.86954646759091444473031262847, 6.80274783066483389766425361700, 7.22871495038663873112013199388, 8.050882121400679747355536858476
