| L(s) = 1 | − 3.61e10·2-s − 5.00e16·3-s − 1.05e21·4-s + 4.57e24·5-s + 1.80e27·6-s − 2.20e29·7-s + 1.23e32·8-s + 2.50e33·9-s − 1.65e35·10-s + 4.33e36·11-s + 5.28e37·12-s + 3.74e39·13-s + 7.94e39·14-s − 2.28e41·15-s − 1.96e42·16-s + 4.48e43·17-s − 9.03e43·18-s + 3.60e45·19-s − 4.83e45·20-s + 1.10e46·21-s − 1.56e47·22-s + 2.49e48·23-s − 6.17e48·24-s − 2.14e49·25-s − 1.35e50·26-s − 1.25e50·27-s + 2.32e50·28-s + ⋯ |
| L(s) = 1 | − 0.743·2-s − 0.577·3-s − 0.447·4-s + 0.703·5-s + 0.429·6-s − 0.219·7-s + 1.07·8-s + 0.333·9-s − 0.522·10-s + 0.464·11-s + 0.258·12-s + 1.06·13-s + 0.163·14-s − 0.406·15-s − 0.351·16-s + 0.934·17-s − 0.247·18-s + 1.44·19-s − 0.314·20-s + 0.126·21-s − 0.345·22-s + 1.13·23-s − 0.621·24-s − 0.505·25-s − 0.793·26-s − 0.192·27-s + 0.0982·28-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
| \(L(36)\) |
\(\approx\) |
\(1.436274116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436274116\) |
| \(L(\frac{73}{2})\) |
|
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 | \( 1 + 3.61e10T + 2.36e21T^{2} \) |
| 5 | \( 1 - 4.57e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 2.20e29T + 1.00e60T^{2} \) |
| 11 | \( 1 - 4.33e36T + 8.68e73T^{2} \) |
| 13 | \( 1 - 3.74e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 4.48e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 3.60e45T + 6.18e90T^{2} \) |
| 23 | \( 1 - 2.49e48T + 4.81e96T^{2} \) |
| 29 | \( 1 + 4.79e51T + 6.76e103T^{2} \) |
| 31 | \( 1 - 7.02e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 1.51e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 3.49e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 8.05e57T + 9.46e115T^{2} \) |
| 47 | \( 1 + 2.12e59T + 5.23e118T^{2} \) |
| 53 | \( 1 + 2.29e61T + 2.65e122T^{2} \) |
| 59 | \( 1 - 4.78e61T + 5.37e125T^{2} \) |
| 61 | \( 1 + 3.88e62T + 5.73e126T^{2} \) |
| 67 | \( 1 - 8.84e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 2.94e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 1.51e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 2.53e67T + 5.38e134T^{2} \) |
| 83 | \( 1 - 7.66e67T + 1.79e136T^{2} \) |
| 89 | \( 1 - 1.19e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 1.55e70T + 1.15e141T^{2} \) |
| 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
−13.02947761721215649810601915130, −11.34513525412045126252448896524, −9.998371859972523933295106489137, −9.181432676663921642594211517691, −7.68462418300590742152400076574, −6.16627529388454230321080537758, −5.01240694026772534906643973402, −3.46769983108638818664965486778, −1.47478487818280430473540585328, −0.77362978328990032517317182077,
0.77362978328990032517317182077, 1.47478487818280430473540585328, 3.46769983108638818664965486778, 5.01240694026772534906643973402, 6.16627529388454230321080537758, 7.68462418300590742152400076574, 9.181432676663921642594211517691, 9.998371859972523933295106489137, 11.34513525412045126252448896524, 13.02947761721215649810601915130
