| L(s) = 1 | + 2-s + 3.11·3-s + 4-s + 3.52·5-s + 3.11·6-s + 1.63·7-s + 8-s + 6.69·9-s + 3.52·10-s − 1.92·11-s + 3.11·12-s − 4.26·13-s + 1.63·14-s + 10.9·15-s + 16-s − 0.309·17-s + 6.69·18-s − 19-s + 3.52·20-s + 5.08·21-s − 1.92·22-s + 7.57·23-s + 3.11·24-s + 7.45·25-s − 4.26·26-s + 11.4·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.79·3-s + 0.5·4-s + 1.57·5-s + 1.27·6-s + 0.617·7-s + 0.353·8-s + 2.23·9-s + 1.11·10-s − 0.580·11-s + 0.898·12-s − 1.18·13-s + 0.436·14-s + 2.83·15-s + 0.250·16-s − 0.0749·17-s + 1.57·18-s − 0.229·19-s + 0.789·20-s + 1.10·21-s − 0.410·22-s + 1.57·23-s + 0.635·24-s + 1.49·25-s − 0.836·26-s + 2.21·27-s + 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.260499038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.260499038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 0.309T + 17T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 0.310T + 31T^{2} \) |
| 37 | \( 1 + 2.33T + 37T^{2} \) |
| 41 | \( 1 - 0.172T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 + 5.27T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 0.906T + 97T^{2} \) |
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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
−7.63639841211409435195448080012, −7.32834634389329739712748358181, −6.54131436849143635499039400907, −5.50225197601856877120868494279, −5.02680011252260892775833424195, −4.28940225497498697870896543373, −3.24563223675261380105122638950, −2.62876519180677309435629660644, −2.09365951667412626839075917614, −1.46672622725348377443502697480,
1.46672622725348377443502697480, 2.09365951667412626839075917614, 2.62876519180677309435629660644, 3.24563223675261380105122638950, 4.28940225497498697870896543373, 5.02680011252260892775833424195, 5.50225197601856877120868494279, 6.54131436849143635499039400907, 7.32834634389329739712748358181, 7.63639841211409435195448080012
