| L(s) = 1 | − 6.24·3-s − 25·5-s − 49·7-s − 203.·9-s − 89.4·11-s + 459.·13-s + 156.·15-s + 701.·17-s + 396.·19-s + 306.·21-s + 4.31e3·23-s + 625·25-s + 2.79e3·27-s − 1.43e3·29-s − 3.66e3·31-s + 558.·33-s + 1.22e3·35-s − 3.16e3·37-s − 2.86e3·39-s − 6.35e3·41-s + 1.55e4·43-s + 5.09e3·45-s + 3.99e3·47-s + 2.40e3·49-s − 4.38e3·51-s − 2.44e4·53-s + 2.23e3·55-s + ⋯ |
| L(s) = 1 | − 0.400·3-s − 0.447·5-s − 0.377·7-s − 0.839·9-s − 0.222·11-s + 0.753·13-s + 0.179·15-s + 0.588·17-s + 0.251·19-s + 0.151·21-s + 1.70·23-s + 0.200·25-s + 0.737·27-s − 0.317·29-s − 0.685·31-s + 0.0893·33-s + 0.169·35-s − 0.380·37-s − 0.302·39-s − 0.590·41-s + 1.27·43-s + 0.375·45-s + 0.263·47-s + 0.142·49-s − 0.235·51-s − 1.19·53-s + 0.0996·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 6.24T + 243T^{2} \) |
| 11 | \( 1 + 89.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 459.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 701.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 396.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.66e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.55e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.99e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.39e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 853.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.37e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−9.437545750775655891767029553858, −8.705181032959064670184164425361, −7.74413190142952824025696488517, −6.78251490105059860401189540159, −5.80237491974682400045490602115, −5.01225265515751654584688826594, −3.65798524363005277916745848626, −2.82655552320397336928600724721, −1.13744468595766172569024831670, 0,
1.13744468595766172569024831670, 2.82655552320397336928600724721, 3.65798524363005277916745848626, 5.01225265515751654584688826594, 5.80237491974682400045490602115, 6.78251490105059860401189540159, 7.74413190142952824025696488517, 8.705181032959064670184164425361, 9.437545750775655891767029553858
