| L(s) = 1 | + 2.19e12·2-s − 8.12e19·3-s + 4.83e24·4-s − 9.96e28·5-s − 1.78e32·6-s − 5.66e34·7-s + 1.06e37·8-s + 2.60e39·9-s − 2.19e41·10-s − 1.84e42·11-s − 3.92e44·12-s + 2.57e46·13-s − 1.24e47·14-s + 8.09e48·15-s + 2.33e49·16-s + 6.69e50·17-s + 5.72e51·18-s + 7.59e52·19-s − 4.81e53·20-s + 4.59e54·21-s − 4.05e54·22-s − 4.69e56·23-s − 8.63e56·24-s − 4.13e56·25-s + 5.66e58·26-s + 1.12e59·27-s − 2.73e59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.28·3-s + 0.5·4-s − 0.979·5-s − 0.908·6-s − 0.480·7-s + 0.353·8-s + 0.652·9-s − 0.692·10-s − 0.111·11-s − 0.642·12-s + 1.52·13-s − 0.339·14-s + 1.25·15-s + 0.250·16-s + 0.578·17-s + 0.461·18-s + 0.648·19-s − 0.489·20-s + 0.617·21-s − 0.0789·22-s − 1.44·23-s − 0.454·24-s − 0.0399·25-s + 1.07·26-s + 0.446·27-s − 0.240·28-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 | \( 1 + 8.12e19T + 3.99e39T^{2} \) |
| 5 | \( 1 + 9.96e28T + 1.03e58T^{2} \) |
| 7 | \( 1 + 5.66e34T + 1.39e70T^{2} \) |
| 11 | \( 1 + 1.84e42T + 2.72e86T^{2} \) |
| 13 | \( 1 - 2.57e46T + 2.86e92T^{2} \) |
| 17 | \( 1 - 6.69e50T + 1.34e102T^{2} \) |
| 19 | \( 1 - 7.59e52T + 1.36e106T^{2} \) |
| 23 | \( 1 + 4.69e56T + 1.05e113T^{2} \) |
| 29 | \( 1 - 2.67e60T + 2.39e121T^{2} \) |
| 31 | \( 1 - 1.41e62T + 6.06e123T^{2} \) |
| 37 | \( 1 + 1.78e65T + 1.44e130T^{2} \) |
| 41 | \( 1 - 4.67e66T + 7.26e133T^{2} \) |
| 43 | \( 1 - 3.40e67T + 3.78e135T^{2} \) |
| 47 | \( 1 + 2.61e69T + 6.08e138T^{2} \) |
| 53 | \( 1 - 1.87e71T + 1.30e143T^{2} \) |
| 59 | \( 1 - 4.41e73T + 9.56e146T^{2} \) |
| 61 | \( 1 + 2.05e74T + 1.52e148T^{2} \) |
| 67 | \( 1 + 1.07e76T + 3.66e151T^{2} \) |
| 71 | \( 1 - 4.83e76T + 4.51e153T^{2} \) |
| 73 | \( 1 + 1.47e77T + 4.52e154T^{2} \) |
| 79 | \( 1 - 6.65e78T + 3.18e157T^{2} \) |
| 83 | \( 1 + 3.99e79T + 1.92e159T^{2} \) |
| 89 | \( 1 + 5.21e80T + 6.30e161T^{2} \) |
| 97 | \( 1 - 1.44e82T + 7.98e164T^{2} \) |
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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
−12.21548048033943306664942386842, −11.55660556469340169523783432253, −10.34335452321249936868976268539, −8.090251270909767711863743269991, −6.54684140476221626789495529940, −5.66935325642083071104520527501, −4.30460381187871454343949550764, −3.23658602677706081404244774037, −1.16124333277531376393655580259, 0,
1.16124333277531376393655580259, 3.23658602677706081404244774037, 4.30460381187871454343949550764, 5.66935325642083071104520527501, 6.54684140476221626789495529940, 8.090251270909767711863743269991, 10.34335452321249936868976268539, 11.55660556469340169523783432253, 12.21548048033943306664942386842
