| L(s) = 1 | + 16.2·3-s − 69.5·5-s − 49·7-s + 21.5·9-s + 202.·11-s − 112.·13-s − 1.13e3·15-s + 1.31e3·17-s + 35.9·19-s − 796.·21-s − 1.73e3·23-s + 1.71e3·25-s − 3.60e3·27-s − 710.·29-s − 185.·31-s + 3.29e3·33-s + 3.40e3·35-s + 6.19e3·37-s − 1.82e3·39-s + 3.56e3·41-s + 2.29e3·43-s − 1.49e3·45-s − 1.14e4·47-s + 2.40e3·49-s + 2.14e4·51-s + 3.57e4·53-s − 1.40e4·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s − 1.24·5-s − 0.377·7-s + 0.0887·9-s + 0.504·11-s − 0.184·13-s − 1.29·15-s + 1.10·17-s + 0.0228·19-s − 0.394·21-s − 0.683·23-s + 0.547·25-s − 0.950·27-s − 0.156·29-s − 0.0347·31-s + 0.526·33-s + 0.470·35-s + 0.744·37-s − 0.192·39-s + 0.331·41-s + 0.189·43-s − 0.110·45-s − 0.756·47-s + 0.142·49-s + 1.15·51-s + 1.74·53-s − 0.627·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.074394136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074394136\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 16.2T + 243T^{2} \) |
| 5 | \( 1 + 69.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 202.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 112.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.31e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 35.9T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 710.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 185.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.11e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.04e4T + 8.58e9T^{2} \) |
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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
−10.08803311902992730309281390406, −9.301365864763241107784134233619, −8.295867319253681322437039612349, −7.81889983793071980388532924942, −6.85064128075560694617932683989, −5.51889833391418515002995387430, −4.00669748212604145906264059436, −3.50752497688558667960623513053, −2.35315084409597991929160981913, −0.69496106859172722761191912365,
0.69496106859172722761191912365, 2.35315084409597991929160981913, 3.50752497688558667960623513053, 4.00669748212604145906264059436, 5.51889833391418515002995387430, 6.85064128075560694617932683989, 7.81889983793071980388532924942, 8.295867319253681322437039612349, 9.301365864763241107784134233619, 10.08803311902992730309281390406
