| L(s) = 1 | + 2.39·2-s + 3.74·4-s + 0.714·5-s − 0.642·7-s + 4.17·8-s + 1.71·10-s + 11-s + 3.57·13-s − 1.53·14-s + 2.51·16-s + 0.970·17-s − 1.66·19-s + 2.67·20-s + 2.39·22-s + 6.84·23-s − 4.48·25-s + 8.57·26-s − 2.40·28-s − 2.83·29-s + 4.42·31-s − 2.32·32-s + 2.32·34-s − 0.458·35-s + 1.60·37-s − 3.99·38-s + 2.98·40-s + 7.68·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.87·4-s + 0.319·5-s − 0.242·7-s + 1.47·8-s + 0.541·10-s + 0.301·11-s + 0.992·13-s − 0.411·14-s + 0.628·16-s + 0.235·17-s − 0.382·19-s + 0.597·20-s + 0.510·22-s + 1.42·23-s − 0.897·25-s + 1.68·26-s − 0.454·28-s − 0.526·29-s + 0.794·31-s − 0.410·32-s + 0.398·34-s − 0.0775·35-s + 0.263·37-s − 0.647·38-s + 0.471·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.550864469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.550864469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 - 0.714T + 5T^{2} \) |
| 7 | \( 1 + 0.642T + 7T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 - 0.970T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 - 4.23T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 + 3.90T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.96T + 97T^{2} \) |
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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
−7.76572329999779299827567748464, −7.10705807621557125485323462508, −6.24029487362303989310586146072, −5.96383877654938302245324185124, −5.21222179207268033463710390012, −4.32770075599899119540708744210, −3.82512468166379430156991530471, −2.98895503139119476535882162217, −2.25736835054384927207204513118, −1.10448638662444862673540694055,
1.10448638662444862673540694055, 2.25736835054384927207204513118, 2.98895503139119476535882162217, 3.82512468166379430156991530471, 4.32770075599899119540708744210, 5.21222179207268033463710390012, 5.96383877654938302245324185124, 6.24029487362303989310586146072, 7.10705807621557125485323462508, 7.76572329999779299827567748464
