| L(s) = 1 | − 2.19e12·2-s + 2.32e18·3-s + 4.83e24·4-s + 1.63e29·5-s − 5.11e30·6-s − 8.77e33·7-s − 1.06e37·8-s − 3.98e39·9-s − 3.59e41·10-s − 4.41e42·11-s + 1.12e43·12-s − 1.03e45·13-s + 1.93e46·14-s + 3.80e47·15-s + 2.33e49·16-s + 7.89e50·17-s + 8.76e51·18-s + 1.66e53·19-s + 7.91e53·20-s − 2.03e52·21-s + 9.70e54·22-s − 3.15e56·23-s − 2.47e55·24-s + 1.64e58·25-s + 2.27e57·26-s − 1.85e58·27-s − 4.24e58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0367·3-s + 0.5·4-s + 1.60·5-s − 0.0260·6-s − 0.0744·7-s − 0.353·8-s − 0.998·9-s − 1.13·10-s − 0.267·11-s + 0.0183·12-s − 0.0610·13-s + 0.0526·14-s + 0.0592·15-s + 0.250·16-s + 0.681·17-s + 0.706·18-s + 1.42·19-s + 0.804·20-s − 0.00273·21-s + 0.188·22-s − 0.971·23-s − 0.0130·24-s + 1.58·25-s + 0.0431·26-s − 0.0735·27-s − 0.0372·28-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
| \(L(42)\) |
\(\approx\) |
\(2.027545440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027545440\) |
| \(L(\frac{85}{2})\) |
|
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 | \( 1 - 2.32e18T + 3.99e39T^{2} \) |
| 5 | \( 1 - 1.63e29T + 1.03e58T^{2} \) |
| 7 | \( 1 + 8.77e33T + 1.39e70T^{2} \) |
| 11 | \( 1 + 4.41e42T + 2.72e86T^{2} \) |
| 13 | \( 1 + 1.03e45T + 2.86e92T^{2} \) |
| 17 | \( 1 - 7.89e50T + 1.34e102T^{2} \) |
| 19 | \( 1 - 1.66e53T + 1.36e106T^{2} \) |
| 23 | \( 1 + 3.15e56T + 1.05e113T^{2} \) |
| 29 | \( 1 + 7.28e60T + 2.39e121T^{2} \) |
| 31 | \( 1 - 1.29e62T + 6.06e123T^{2} \) |
| 37 | \( 1 - 1.79e65T + 1.44e130T^{2} \) |
| 41 | \( 1 - 3.38e66T + 7.26e133T^{2} \) |
| 43 | \( 1 + 3.48e67T + 3.78e135T^{2} \) |
| 47 | \( 1 + 1.23e69T + 6.08e138T^{2} \) |
| 53 | \( 1 - 3.80e71T + 1.30e143T^{2} \) |
| 59 | \( 1 - 2.39e73T + 9.56e146T^{2} \) |
| 61 | \( 1 + 3.26e73T + 1.52e148T^{2} \) |
| 67 | \( 1 + 5.45e75T + 3.66e151T^{2} \) |
| 71 | \( 1 - 6.95e76T + 4.51e153T^{2} \) |
| 73 | \( 1 - 9.48e76T + 4.52e154T^{2} \) |
| 79 | \( 1 + 8.48e78T + 3.18e157T^{2} \) |
| 83 | \( 1 - 4.88e79T + 1.92e159T^{2} \) |
| 89 | \( 1 - 1.14e81T + 6.30e161T^{2} \) |
| 97 | \( 1 - 5.10e82T + 7.98e164T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.41379496684283279997256823910, −11.64884179152788444134346151978, −10.10692298276767702176028971077, −9.341850961182419423369906951027, −7.894911232942774342096213026871, −6.20535689093710138932044460470, −5.38990677386099909510799920598, −3.02793283270186350080828190482, −2.01471965848111635953369991381, −0.791014437359105890905061053263,
0.791014437359105890905061053263, 2.01471965848111635953369991381, 3.02793283270186350080828190482, 5.38990677386099909510799920598, 6.20535689093710138932044460470, 7.894911232942774342096213026871, 9.341850961182419423369906951027, 10.10692298276767702176028971077, 11.64884179152788444134346151978, 13.41379496684283279997256823910
