| L(s) = 1 | − 1.31·2-s + 2.12·3-s − 30.2·4-s + 25·5-s − 2.80·6-s − 96.2·7-s + 82.0·8-s − 238.·9-s − 32.9·10-s − 121·11-s − 64.2·12-s + 901.·13-s + 126.·14-s + 53.0·15-s + 860.·16-s + 924.·17-s + 314.·18-s − 361·19-s − 756.·20-s − 204.·21-s + 159.·22-s + 3.25e3·23-s + 174.·24-s + 625·25-s − 1.18e3·26-s − 1.02e3·27-s + 2.91e3·28-s + ⋯ |
| L(s) = 1 | − 0.233·2-s + 0.136·3-s − 0.945·4-s + 0.447·5-s − 0.0317·6-s − 0.742·7-s + 0.453·8-s − 0.981·9-s − 0.104·10-s − 0.301·11-s − 0.128·12-s + 1.47·13-s + 0.173·14-s + 0.0609·15-s + 0.839·16-s + 0.775·17-s + 0.228·18-s − 0.229·19-s − 0.422·20-s − 0.101·21-s + 0.0702·22-s + 1.28·23-s + 0.0617·24-s + 0.200·25-s − 0.344·26-s − 0.269·27-s + 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.149756209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.149756209\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 1.31T + 32T^{2} \) |
| 3 | \( 1 - 2.12T + 243T^{2} \) |
| 7 | \( 1 + 96.2T + 1.68e4T^{2} \) |
| 13 | \( 1 - 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 924.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.55e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.93e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.33e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.07e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e4T + 8.58e9T^{2} \) |
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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
−9.167513612620576063387089023778, −8.544997761594562384718972374562, −7.77966984783660496514878005797, −6.53766632866469764499723822538, −5.71881525455583523088854587141, −5.04580687760757472673227006380, −3.64299370079448077514670732418, −3.14520568738104294241119623834, −1.61309967391384676713926893054, −0.49233670483556793899574964763,
0.49233670483556793899574964763, 1.61309967391384676713926893054, 3.14520568738104294241119623834, 3.64299370079448077514670732418, 5.04580687760757472673227006380, 5.71881525455583523088854587141, 6.53766632866469764499723822538, 7.77966984783660496514878005797, 8.544997761594562384718972374562, 9.167513612620576063387089023778
