| L(s) = 1 | − 5.95·3-s − 11.6·5-s + 49·7-s − 207.·9-s − 12.4·11-s + 8.71·13-s + 69.5·15-s + 2.12e3·17-s − 2.22e3·19-s − 291.·21-s − 2.94e3·23-s − 2.98e3·25-s + 2.68e3·27-s − 2.46e3·29-s + 9.13e3·31-s + 73.8·33-s − 572.·35-s − 6.68e3·37-s − 51.8·39-s + 1.13e4·41-s + 1.18e4·43-s + 2.42e3·45-s − 4.12e3·47-s + 2.40e3·49-s − 1.26e4·51-s − 3.68e4·53-s + 145.·55-s + ⋯ |
| L(s) = 1 | − 0.381·3-s − 0.208·5-s + 0.377·7-s − 0.854·9-s − 0.0309·11-s + 0.0143·13-s + 0.0798·15-s + 1.78·17-s − 1.41·19-s − 0.144·21-s − 1.16·23-s − 0.956·25-s + 0.708·27-s − 0.544·29-s + 1.70·31-s + 0.0118·33-s − 0.0789·35-s − 0.802·37-s − 0.00546·39-s + 1.05·41-s + 0.974·43-s + 0.178·45-s − 0.272·47-s + 0.142·49-s − 0.681·51-s − 1.80·53-s + 0.00646·55-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
| \(L(3)\) |
\(\approx\) |
\(1.361911839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361911839\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.95T + 243T^{2} \) |
| 5 | \( 1 + 11.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 12.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 8.71T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.74e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.08e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.86e4T + 8.58e9T^{2} \) |
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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
−10.39977750552413925763207042882, −9.470772449393220427973340834918, −8.228565505434590254280856388620, −7.81439725177778224660677489487, −6.32769957628878377850759055843, −5.68303207894274139084350573998, −4.54515854753738174453378038130, −3.39606664320655296219989868720, −2.07091003260562595982557923305, −0.60234019061133957939105333859,
0.60234019061133957939105333859, 2.07091003260562595982557923305, 3.39606664320655296219989868720, 4.54515854753738174453378038130, 5.68303207894274139084350573998, 6.32769957628878377850759055843, 7.81439725177778224660677489487, 8.228565505434590254280856388620, 9.470772449393220427973340834918, 10.39977750552413925763207042882
