| L(s) = 1 | − 7.41e10·2-s − 5.00e16·3-s + 3.14e21·4-s − 1.28e25·5-s + 3.71e27·6-s − 1.80e28·7-s − 5.78e31·8-s + 2.50e33·9-s + 9.52e35·10-s + 9.59e36·11-s − 1.57e38·12-s − 3.58e39·13-s + 1.33e39·14-s + 6.42e41·15-s − 3.12e42·16-s + 3.97e43·17-s − 1.85e44·18-s + 1.44e45·19-s − 4.03e46·20-s + 9.02e44·21-s − 7.11e47·22-s − 7.47e46·23-s + 2.89e48·24-s + 1.22e50·25-s + 2.65e50·26-s − 1.25e50·27-s − 5.66e49·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.577·3-s + 1.33·4-s − 1.97·5-s + 0.881·6-s − 0.0179·7-s − 0.503·8-s + 0.333·9-s + 3.01·10-s + 1.02·11-s − 0.767·12-s − 1.02·13-s + 0.0274·14-s + 1.13·15-s − 0.560·16-s + 0.828·17-s − 0.508·18-s + 0.579·19-s − 2.62·20-s + 0.0103·21-s − 1.57·22-s − 0.0340·23-s + 0.290·24-s + 2.89·25-s + 1.55·26-s − 0.192·27-s − 0.0239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(\approx\) |
\(0.2441277138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2441277138\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.00e16T \) |
| good | 2 | \( 1 + 7.41e10T + 2.36e21T^{2} \) |
| 5 | \( 1 + 1.28e25T + 4.23e49T^{2} \) |
| 7 | \( 1 + 1.80e28T + 1.00e60T^{2} \) |
| 11 | \( 1 - 9.59e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 3.58e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 3.97e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 1.44e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 7.47e46T + 4.81e96T^{2} \) |
| 29 | \( 1 - 8.56e51T + 6.76e103T^{2} \) |
| 31 | \( 1 + 1.40e53T + 7.70e105T^{2} \) |
| 37 | \( 1 + 1.84e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 1.60e57T + 3.21e114T^{2} \) |
| 43 | \( 1 + 1.56e58T + 9.46e115T^{2} \) |
| 47 | \( 1 - 7.88e57T + 5.23e118T^{2} \) |
| 53 | \( 1 - 2.82e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 1.10e63T + 5.37e125T^{2} \) |
| 61 | \( 1 - 7.22e62T + 5.73e126T^{2} \) |
| 67 | \( 1 + 3.01e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 5.43e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 2.36e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 1.15e67T + 5.38e134T^{2} \) |
| 83 | \( 1 - 1.12e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 1.53e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 3.69e70T + 1.15e141T^{2} \) |
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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
−12.11362693575805959211098547648, −11.52473294275427251240740202044, −10.21059933674758623267816093057, −8.797775454753550392459490275806, −7.63575855421538342719267704684, −6.93636882620632954772168594870, −4.72253590940922765366888114004, −3.38554772555346026622260761978, −1.36728116652809459890841653460, −0.34656249697506813628412481046,
0.34656249697506813628412481046, 1.36728116652809459890841653460, 3.38554772555346026622260761978, 4.72253590940922765366888114004, 6.93636882620632954772168594870, 7.63575855421538342719267704684, 8.797775454753550392459490275806, 10.21059933674758623267816093057, 11.52473294275427251240740202044, 12.11362693575805959211098547648
