| L(s) = 1 | + 1.63e14·2-s + 6.46e21·3-s − 1.27e28·4-s + 1.20e33·5-s + 1.06e36·6-s − 2.14e40·7-s − 8.58e42·8-s − 2.07e45·9-s + 1.96e47·10-s + 1.70e49·11-s − 8.22e49·12-s + 1.01e53·13-s − 3.52e54·14-s + 7.76e54·15-s − 9.03e56·16-s + 2.27e58·17-s − 3.40e59·18-s + 5.33e60·19-s − 1.52e61·20-s − 1.38e62·21-s + 2.79e63·22-s + 5.74e62·23-s − 5.55e64·24-s − 1.08e66·25-s + 1.65e67·26-s − 2.71e67·27-s + 2.73e68·28-s + ⋯ |
| L(s) = 1 | + 0.823·2-s + 0.140·3-s − 0.321·4-s + 0.755·5-s + 0.115·6-s − 1.54·7-s − 1.08·8-s − 0.980·9-s + 0.622·10-s + 0.582·11-s − 0.0451·12-s + 1.23·13-s − 1.27·14-s + 0.106·15-s − 0.575·16-s + 0.812·17-s − 0.807·18-s + 0.968·19-s − 0.242·20-s − 0.217·21-s + 0.479·22-s + 0.0119·23-s − 0.152·24-s − 0.429·25-s + 1.02·26-s − 0.278·27-s + 0.497·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(2.601841532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601841532\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.63e14T + 3.96e28T^{2} \) |
| 3 | \( 1 - 6.46e21T + 2.12e45T^{2} \) |
| 5 | \( 1 - 1.20e33T + 2.52e66T^{2} \) |
| 7 | \( 1 + 2.14e40T + 1.92e80T^{2} \) |
| 11 | \( 1 - 1.70e49T + 8.55e98T^{2} \) |
| 13 | \( 1 - 1.01e53T + 6.67e105T^{2} \) |
| 17 | \( 1 - 2.27e58T + 7.80e116T^{2} \) |
| 19 | \( 1 - 5.33e60T + 3.03e121T^{2} \) |
| 23 | \( 1 - 5.74e62T + 2.31e129T^{2} \) |
| 29 | \( 1 - 2.72e69T + 8.46e138T^{2} \) |
| 31 | \( 1 - 8.24e70T + 4.77e141T^{2} \) |
| 37 | \( 1 + 5.13e74T + 9.53e148T^{2} \) |
| 41 | \( 1 - 7.35e76T + 1.63e153T^{2} \) |
| 43 | \( 1 + 6.17e76T + 1.51e155T^{2} \) |
| 47 | \( 1 - 3.75e78T + 7.06e158T^{2} \) |
| 53 | \( 1 - 2.90e81T + 6.40e163T^{2} \) |
| 59 | \( 1 - 1.68e84T + 1.70e168T^{2} \) |
| 61 | \( 1 - 2.77e84T + 4.03e169T^{2} \) |
| 67 | \( 1 - 3.14e85T + 2.99e173T^{2} \) |
| 71 | \( 1 + 4.28e87T + 7.40e175T^{2} \) |
| 73 | \( 1 + 4.92e88T + 1.03e177T^{2} \) |
| 79 | \( 1 - 2.67e90T + 1.88e180T^{2} \) |
| 83 | \( 1 + 1.35e91T + 2.05e182T^{2} \) |
| 89 | \( 1 - 3.19e91T + 1.55e185T^{2} \) |
| 97 | \( 1 - 2.36e94T + 5.53e188T^{2} \) |
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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
−14.13745852914345984276940564387, −13.42097589598463524550998302522, −12.02382788658162824266888615956, −9.857912674978898069722511781363, −8.793103259441492987936984616374, −6.33421536558988931321784485841, −5.61121454848932533560555340743, −3.69397197032653277173058035153, −2.87561420806360304425743749642, −0.77616404857505558687246745598,
0.77616404857505558687246745598, 2.87561420806360304425743749642, 3.69397197032653277173058035153, 5.61121454848932533560555340743, 6.33421536558988931321784485841, 8.793103259441492987936984616374, 9.857912674978898069722511781363, 12.02382788658162824266888615956, 13.42097589598463524550998302522, 14.13745852914345984276940564387
