| L(s) = 1 | − 2.66e10·2-s + 5.00e16·3-s − 1.65e21·4-s + 1.17e25·5-s − 1.33e27·6-s − 5.64e29·7-s + 1.06e32·8-s + 2.50e33·9-s − 3.12e35·10-s + 8.53e36·11-s − 8.25e37·12-s − 4.54e39·13-s + 1.50e40·14-s + 5.86e41·15-s + 1.04e42·16-s − 1.14e43·17-s − 6.67e43·18-s − 2.21e45·19-s − 1.93e46·20-s − 2.82e46·21-s − 2.27e47·22-s − 3.03e48·23-s + 5.35e48·24-s + 9.50e49·25-s + 1.21e50·26-s + 1.25e50·27-s + 9.31e50·28-s + ⋯ |
| L(s) = 1 | − 0.548·2-s + 0.577·3-s − 0.699·4-s + 1.80·5-s − 0.316·6-s − 0.563·7-s + 0.932·8-s + 0.333·9-s − 0.987·10-s + 0.915·11-s − 0.403·12-s − 1.29·13-s + 0.308·14-s + 1.03·15-s + 0.187·16-s − 0.238·17-s − 0.182·18-s − 0.891·19-s − 1.25·20-s − 0.325·21-s − 0.502·22-s − 1.38·23-s + 0.538·24-s + 2.24·25-s + 0.710·26-s + 0.192·27-s + 0.393·28-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
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
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 | \( 1 + 2.66e10T + 2.36e21T^{2} \) |
| 5 | \( 1 - 1.17e25T + 4.23e49T^{2} \) |
| 7 | \( 1 + 5.64e29T + 1.00e60T^{2} \) |
| 11 | \( 1 - 8.53e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 4.54e39T + 1.23e79T^{2} \) |
| 17 | \( 1 + 1.14e43T + 2.30e87T^{2} \) |
| 19 | \( 1 + 2.21e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 3.03e48T + 4.81e96T^{2} \) |
| 29 | \( 1 + 8.42e51T + 6.76e103T^{2} \) |
| 31 | \( 1 - 7.18e51T + 7.70e105T^{2} \) |
| 37 | \( 1 + 2.81e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 1.27e57T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.83e57T + 9.46e115T^{2} \) |
| 47 | \( 1 - 2.64e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 2.81e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 5.46e62T + 5.37e125T^{2} \) |
| 61 | \( 1 - 2.38e63T + 5.73e126T^{2} \) |
| 67 | \( 1 - 1.16e65T + 4.48e129T^{2} \) |
| 71 | \( 1 + 6.39e64T + 2.75e131T^{2} \) |
| 73 | \( 1 + 1.87e66T + 1.97e132T^{2} \) |
| 79 | \( 1 + 2.89e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 2.14e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 1.37e69T + 2.55e138T^{2} \) |
| 97 | \( 1 + 6.37e70T + 1.15e141T^{2} \) |
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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
−12.68095827276484047248499543014, −10.15325473360599731901522716051, −9.605343645672633918490536178050, −8.683890129578603754765857598671, −6.92630184088573804415546056816, −5.57406591898591413653151581390, −4.11639679509709360265947264357, −2.38994095500178699802708786479, −1.52268485849087914547342268614, 0,
1.52268485849087914547342268614, 2.38994095500178699802708786479, 4.11639679509709360265947264357, 5.57406591898591413653151581390, 6.92630184088573804415546056816, 8.683890129578603754765857598671, 9.605343645672633918490536178050, 10.15325473360599731901522716051, 12.68095827276484047248499543014
