| L(s) = 1 | + 2.19e12·2-s + 8.79e19·3-s + 4.83e24·4-s − 8.42e28·5-s + 1.93e32·6-s + 1.88e34·7-s + 1.06e37·8-s + 3.73e39·9-s − 1.85e41·10-s − 2.30e43·11-s + 4.25e44·12-s − 1.10e46·13-s + 4.14e46·14-s − 7.40e48·15-s + 2.33e49·16-s − 6.53e50·17-s + 8.21e51·18-s + 1.88e53·19-s − 4.07e53·20-s + 1.65e54·21-s − 5.05e55·22-s − 3.40e56·23-s + 9.34e56·24-s − 3.24e57·25-s − 2.43e58·26-s − 2.24e58·27-s + 9.11e58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s − 0.828·5-s + 0.983·6-s + 0.159·7-s + 0.353·8-s + 0.936·9-s − 0.585·10-s − 1.39·11-s + 0.695·12-s − 0.653·13-s + 0.113·14-s − 1.15·15-s + 0.250·16-s − 0.564·17-s + 0.661·18-s + 1.61·19-s − 0.414·20-s + 0.222·21-s − 0.985·22-s − 1.04·23-s + 0.491·24-s − 0.314·25-s − 0.461·26-s − 0.0889·27-s + 0.0799·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(84-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+83/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(42)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{85}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.19e12T \) |
| good | 3 | \( 1 - 8.79e19T + 3.99e39T^{2} \) |
| 5 | \( 1 + 8.42e28T + 1.03e58T^{2} \) |
| 7 | \( 1 - 1.88e34T + 1.39e70T^{2} \) |
| 11 | \( 1 + 2.30e43T + 2.72e86T^{2} \) |
| 13 | \( 1 + 1.10e46T + 2.86e92T^{2} \) |
| 17 | \( 1 + 6.53e50T + 1.34e102T^{2} \) |
| 19 | \( 1 - 1.88e53T + 1.36e106T^{2} \) |
| 23 | \( 1 + 3.40e56T + 1.05e113T^{2} \) |
| 29 | \( 1 + 3.75e60T + 2.39e121T^{2} \) |
| 31 | \( 1 + 1.06e62T + 6.06e123T^{2} \) |
| 37 | \( 1 - 1.36e65T + 1.44e130T^{2} \) |
| 41 | \( 1 - 2.40e66T + 7.26e133T^{2} \) |
| 43 | \( 1 + 9.96e67T + 3.78e135T^{2} \) |
| 47 | \( 1 - 2.73e69T + 6.08e138T^{2} \) |
| 53 | \( 1 + 2.76e71T + 1.30e143T^{2} \) |
| 59 | \( 1 - 9.63e72T + 9.56e146T^{2} \) |
| 61 | \( 1 + 2.93e73T + 1.52e148T^{2} \) |
| 67 | \( 1 - 9.23e75T + 3.66e151T^{2} \) |
| 71 | \( 1 - 9.92e76T + 4.51e153T^{2} \) |
| 73 | \( 1 + 1.74e77T + 4.52e154T^{2} \) |
| 79 | \( 1 + 9.46e78T + 3.18e157T^{2} \) |
| 83 | \( 1 - 6.81e79T + 1.92e159T^{2} \) |
| 89 | \( 1 + 1.00e81T + 6.30e161T^{2} \) |
| 97 | \( 1 + 3.80e82T + 7.98e164T^{2} \) |
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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
−12.89640441482290217013362899042, −11.40541533912570572109262997958, −9.689406765552795149344359713381, −8.032263098295527705705042657156, −7.44771454441060221788640593271, −5.30824082414509021481060633407, −3.94380373597255995856345700230, −2.97046395665437885529529399452, −1.99256674234458973768592276561, 0,
1.99256674234458973768592276561, 2.97046395665437885529529399452, 3.94380373597255995856345700230, 5.30824082414509021481060633407, 7.44771454441060221788640593271, 8.032263098295527705705042657156, 9.689406765552795149344359713381, 11.40541533912570572109262997958, 12.89640441482290217013362899042
