| L(s) = 1 | + 4.39e10·2-s − 5.00e16·3-s − 4.26e20·4-s + 4.61e24·5-s − 2.20e27·6-s − 1.72e30·7-s − 1.22e32·8-s + 2.50e33·9-s + 2.02e35·10-s − 3.29e36·11-s + 2.13e37·12-s − 5.27e39·13-s − 7.60e40·14-s − 2.30e41·15-s − 4.38e42·16-s + 2.19e42·17-s + 1.10e44·18-s + 1.32e45·19-s − 1.96e45·20-s + 8.65e46·21-s − 1.45e47·22-s + 4.12e47·23-s + 6.13e48·24-s − 2.10e49·25-s − 2.32e50·26-s − 1.25e50·27-s + 7.37e50·28-s + ⋯ |
| L(s) = 1 | + 0.905·2-s − 0.577·3-s − 0.180·4-s + 0.708·5-s − 0.522·6-s − 1.72·7-s − 1.06·8-s + 0.333·9-s + 0.641·10-s − 0.354·11-s + 0.104·12-s − 1.50·13-s − 1.56·14-s − 0.409·15-s − 0.786·16-s + 0.0457·17-s + 0.301·18-s + 0.534·19-s − 0.127·20-s + 0.996·21-s − 0.320·22-s + 0.187·23-s + 0.617·24-s − 0.498·25-s − 1.36·26-s − 0.192·27-s + 0.311·28-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
| \(L(36)\) |
\(\approx\) |
\(0.9557758314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9557758314\) |
| \(L(\frac{73}{2})\) |
|
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 | \( 1 - 4.39e10T + 2.36e21T^{2} \) |
| 5 | \( 1 - 4.61e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 1.72e30T + 1.00e60T^{2} \) |
| 11 | \( 1 + 3.29e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 5.27e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 2.19e42T + 2.30e87T^{2} \) |
| 19 | \( 1 - 1.32e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 4.12e47T + 4.81e96T^{2} \) |
| 29 | \( 1 - 9.84e51T + 6.76e103T^{2} \) |
| 31 | \( 1 + 1.36e53T + 7.70e105T^{2} \) |
| 37 | \( 1 + 3.72e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 1.73e57T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.55e58T + 9.46e115T^{2} \) |
| 47 | \( 1 + 2.62e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 3.04e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 6.79e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 6.01e62T + 5.73e126T^{2} \) |
| 67 | \( 1 + 5.52e63T + 4.48e129T^{2} \) |
| 71 | \( 1 - 7.86e65T + 2.75e131T^{2} \) |
| 73 | \( 1 - 1.87e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 1.89e66T + 5.38e134T^{2} \) |
| 83 | \( 1 + 1.25e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 3.41e67T + 2.55e138T^{2} \) |
| 97 | \( 1 - 1.31e70T + 1.15e141T^{2} \) |
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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
−12.91851732396526773660213852777, −12.18346260629873526799843742192, −10.10328595660512514502244404141, −9.336807664932400119348588167960, −6.98738392396376823941945051611, −5.88813646318533983875852148696, −5.01811257578576324109465670169, −3.54217760328700585500309314585, −2.45545869526564207663071948102, −0.40315946812409343359648102299,
0.40315946812409343359648102299, 2.45545869526564207663071948102, 3.54217760328700585500309314585, 5.01811257578576324109465670169, 5.88813646318533983875852148696, 6.98738392396376823941945051611, 9.336807664932400119348588167960, 10.10328595660512514502244404141, 12.18346260629873526799843742192, 12.91851732396526773660213852777
