| L(s) = 1 | + 20.5·3-s + 53.3·5-s + 49·7-s + 178.·9-s − 90.7·11-s − 317.·13-s + 1.09e3·15-s + 797.·17-s − 264.·19-s + 1.00e3·21-s + 3.14e3·23-s − 282.·25-s − 1.32e3·27-s + 5.40e3·29-s + 6.73e3·31-s − 1.86e3·33-s + 2.61e3·35-s + 1.56e4·37-s − 6.51e3·39-s + 1.37e3·41-s + 2.26e3·43-s + 9.51e3·45-s − 1.17e4·47-s + 2.40e3·49-s + 1.63e4·51-s + 4.28e3·53-s − 4.83e3·55-s + ⋯ |
| L(s) = 1 | + 1.31·3-s + 0.953·5-s + 0.377·7-s + 0.734·9-s − 0.226·11-s − 0.521·13-s + 1.25·15-s + 0.669·17-s − 0.167·19-s + 0.497·21-s + 1.23·23-s − 0.0904·25-s − 0.350·27-s + 1.19·29-s + 1.25·31-s − 0.297·33-s + 0.360·35-s + 1.88·37-s − 0.686·39-s + 0.128·41-s + 0.186·43-s + 0.700·45-s − 0.773·47-s + 0.142·49-s + 0.881·51-s + 0.209·53-s − 0.215·55-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
| \(L(3)\) |
\(\approx\) |
\(4.603057899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.603057899\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 20.5T + 243T^{2} \) |
| 5 | \( 1 - 53.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 90.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 317.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 797.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 264.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.56e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.17e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.28e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.10e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e5T + 8.58e9T^{2} \) |
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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
−9.908690294168543470476573075191, −9.493160093462103732102492672585, −8.433550881737126810721696864828, −7.82680440928431192352499657250, −6.67333886233058119348918885380, −5.50808277975275293514097333786, −4.42401161635522747180593744831, −2.99061347051715991516813600724, −2.35234410326593723908535343491, −1.11558190798698278568901500262,
1.11558190798698278568901500262, 2.35234410326593723908535343491, 2.99061347051715991516813600724, 4.42401161635522747180593744831, 5.50808277975275293514097333786, 6.67333886233058119348918885380, 7.82680440928431192352499657250, 8.433550881737126810721696864828, 9.493160093462103732102492672585, 9.908690294168543470476573075191
