| L(s) = 1 | + 2.19e12·2-s − 2.35e19·3-s + 4.83e24·4-s + 9.47e28·5-s − 5.18e31·6-s + 1.10e35·7-s + 1.06e37·8-s − 3.43e39·9-s + 2.08e41·10-s − 1.28e41·11-s − 1.13e44·12-s − 2.80e46·13-s + 2.43e47·14-s − 2.23e48·15-s + 2.33e49·16-s + 1.33e50·17-s − 7.55e51·18-s − 2.09e53·19-s + 4.58e53·20-s − 2.60e54·21-s − 2.83e53·22-s + 4.69e54·23-s − 2.50e56·24-s − 1.36e57·25-s − 6.17e58·26-s + 1.74e59·27-s + 5.35e59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.372·3-s + 0.5·4-s + 0.931·5-s − 0.263·6-s + 0.938·7-s + 0.353·8-s − 0.860·9-s + 0.658·10-s − 0.00780·11-s − 0.186·12-s − 1.65·13-s + 0.663·14-s − 0.347·15-s + 0.250·16-s + 0.115·17-s − 0.608·18-s − 1.78·19-s + 0.465·20-s − 0.350·21-s − 0.00551·22-s + 0.0144·23-s − 0.131·24-s − 0.132·25-s − 1.17·26-s + 0.694·27-s + 0.469·28-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 | \( 1 + 2.35e19T + 3.99e39T^{2} \) |
| 5 | \( 1 - 9.47e28T + 1.03e58T^{2} \) |
| 7 | \( 1 - 1.10e35T + 1.39e70T^{2} \) |
| 11 | \( 1 + 1.28e41T + 2.72e86T^{2} \) |
| 13 | \( 1 + 2.80e46T + 2.86e92T^{2} \) |
| 17 | \( 1 - 1.33e50T + 1.34e102T^{2} \) |
| 19 | \( 1 + 2.09e53T + 1.36e106T^{2} \) |
| 23 | \( 1 - 4.69e54T + 1.05e113T^{2} \) |
| 29 | \( 1 - 4.88e60T + 2.39e121T^{2} \) |
| 31 | \( 1 + 7.70e60T + 6.06e123T^{2} \) |
| 37 | \( 1 + 1.13e64T + 1.44e130T^{2} \) |
| 41 | \( 1 + 8.70e66T + 7.26e133T^{2} \) |
| 43 | \( 1 - 3.20e67T + 3.78e135T^{2} \) |
| 47 | \( 1 + 4.74e69T + 6.08e138T^{2} \) |
| 53 | \( 1 - 9.11e70T + 1.30e143T^{2} \) |
| 59 | \( 1 + 3.63e73T + 9.56e146T^{2} \) |
| 61 | \( 1 - 1.91e74T + 1.52e148T^{2} \) |
| 67 | \( 1 - 1.61e75T + 3.66e151T^{2} \) |
| 71 | \( 1 - 3.52e76T + 4.51e153T^{2} \) |
| 73 | \( 1 - 5.89e76T + 4.52e154T^{2} \) |
| 79 | \( 1 + 9.22e78T + 3.18e157T^{2} \) |
| 83 | \( 1 + 2.91e79T + 1.92e159T^{2} \) |
| 89 | \( 1 + 1.37e81T + 6.30e161T^{2} \) |
| 97 | \( 1 + 1.38e82T + 7.98e164T^{2} \) |
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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
−12.57537392954674229963225546670, −11.36132349240418467706841778097, −10.06867854839718962962210861328, −8.299585195443666414629834854204, −6.63104524017406507434902453088, −5.44441110505981947053167142034, −4.59005005014278173264112749086, −2.66018552711328777975433605652, −1.77320169177102475624598716027, 0,
1.77320169177102475624598716027, 2.66018552711328777975433605652, 4.59005005014278173264112749086, 5.44441110505981947053167142034, 6.63104524017406507434902453088, 8.299585195443666414629834854204, 10.06867854839718962962210861328, 11.36132349240418467706841778097, 12.57537392954674229963225546670
