| L(s) = 1 | − 26.3·3-s + 97.5·5-s − 49·7-s + 448.·9-s + 406.·11-s + 905.·13-s − 2.56e3·15-s + 359.·17-s + 1.81e3·19-s + 1.28e3·21-s + 2.16e3·23-s + 6.38e3·25-s − 5.41e3·27-s + 4.09e3·29-s − 4.45e3·31-s − 1.06e4·33-s − 4.77e3·35-s − 8.93e3·37-s − 2.38e4·39-s − 3.41e3·41-s − 8.69e3·43-s + 4.37e4·45-s − 1.42e4·47-s + 2.40e3·49-s − 9.46e3·51-s + 1.46e4·53-s + 3.96e4·55-s + ⋯ |
| L(s) = 1 | − 1.68·3-s + 1.74·5-s − 0.377·7-s + 1.84·9-s + 1.01·11-s + 1.48·13-s − 2.94·15-s + 0.301·17-s + 1.15·19-s + 0.637·21-s + 0.852·23-s + 2.04·25-s − 1.42·27-s + 0.903·29-s − 0.831·31-s − 1.70·33-s − 0.659·35-s − 1.07·37-s − 2.50·39-s − 0.317·41-s − 0.717·43-s + 3.22·45-s − 0.937·47-s + 0.142·49-s − 0.509·51-s + 0.714·53-s + 1.76·55-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
| \(L(3)\) |
\(\approx\) |
\(2.121532546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.121532546\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 26.3T + 243T^{2} \) |
| 5 | \( 1 - 97.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 406.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 905.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 359.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.42e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.80e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
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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
−10.35824490871653557204063131149, −9.642333797594491744076953674747, −8.807300170854804333568177762058, −6.90782291772210852074803451635, −6.38555066611836927786076304783, −5.67855508764826129490640752499, −4.99110009073321185303252887932, −3.43498930681517332967113222293, −1.57286552843896914422680995142, −0.931995593441399261942238545215,
0.931995593441399261942238545215, 1.57286552843896914422680995142, 3.43498930681517332967113222293, 4.99110009073321185303252887932, 5.67855508764826129490640752499, 6.38555066611836927786076304783, 6.90782291772210852074803451635, 8.807300170854804333568177762058, 9.642333797594491744076953674747, 10.35824490871653557204063131149
