L(s) = 1 | − 3.02e5i·2-s + (−5.37e7 − 3.83e8i)3-s − 2.28e10·4-s − 4.03e12i·5-s + (−1.16e14 + 1.62e13i)6-s + 4.01e14·7-s − 1.38e16i·8-s + (−1.44e17 + 4.12e16i)9-s − 1.21e18·10-s − 3.25e18i·11-s + (1.22e18 + 8.75e18i)12-s − 1.15e20·13-s − 1.21e20i·14-s + (−1.54e21 + 2.16e20i)15-s − 5.77e21·16-s − 1.35e22i·17-s + ⋯ |
L(s) = 1 | − 1.15i·2-s + (−0.138 − 0.990i)3-s − 0.332·4-s − 1.05i·5-s + (−1.14 + 0.160i)6-s + 0.246·7-s − 0.770i·8-s + (−0.961 + 0.274i)9-s − 1.21·10-s − 0.585i·11-s + (0.0461 + 0.328i)12-s − 1.02·13-s − 0.284i·14-s + (−1.04 + 0.146i)15-s − 1.22·16-s − 0.965i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{37}{2})\) |
\(\approx\) |
\(1.512546049\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.512546049\) |
\(L(19)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.37e7 + 3.83e8i)T \) |
good | 2 | \( 1 + 3.02e5iT - 6.87e10T^{2} \) |
| 5 | \( 1 + 4.03e12iT - 1.45e25T^{2} \) |
| 7 | \( 1 - 4.01e14T + 2.65e30T^{2} \) |
| 11 | \( 1 + 3.25e18iT - 3.09e37T^{2} \) |
| 13 | \( 1 + 1.15e20T + 1.26e40T^{2} \) |
| 17 | \( 1 + 1.35e22iT - 1.97e44T^{2} \) |
| 19 | \( 1 - 1.84e23T + 1.08e46T^{2} \) |
| 23 | \( 1 - 3.27e24iT - 1.05e49T^{2} \) |
| 29 | \( 1 - 2.12e26iT - 4.42e52T^{2} \) |
| 31 | \( 1 - 1.84e25T + 4.88e53T^{2} \) |
| 37 | \( 1 - 3.11e28T + 2.85e56T^{2} \) |
| 41 | \( 1 + 2.06e29iT - 1.14e58T^{2} \) |
| 43 | \( 1 + 1.49e29T + 6.38e58T^{2} \) |
| 47 | \( 1 - 1.35e29iT - 1.56e60T^{2} \) |
| 53 | \( 1 + 1.17e30iT - 1.18e62T^{2} \) |
| 59 | \( 1 - 4.51e31iT - 5.63e63T^{2} \) |
| 61 | \( 1 + 1.40e32T + 1.87e64T^{2} \) |
| 67 | \( 1 - 3.01e32T + 5.47e65T^{2} \) |
| 71 | \( 1 + 1.78e33iT - 4.41e66T^{2} \) |
| 73 | \( 1 + 4.43e33T + 1.20e67T^{2} \) |
| 79 | \( 1 - 1.68e34T + 2.06e68T^{2} \) |
| 83 | \( 1 - 3.28e34iT - 1.22e69T^{2} \) |
| 89 | \( 1 + 2.03e34iT - 1.50e70T^{2} \) |
| 97 | \( 1 + 9.19e35T + 3.34e71T^{2} \) |
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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
−16.40494856870854492432026002902, −13.68152877397768991320297473972, −12.37190811185708290988876130459, −11.45471832575604412872859689377, −9.325630060394530624024058736127, −7.41289101260352769869952448552, −5.20268147775393257326850863668, −2.94698797465139545560106051261, −1.43750552745232256077835493214, −0.52958644252986728793728414214,
2.73301557727581624006663229202, 4.75557280413119842863514083162, 6.28990950478170886453390461502, 7.79277642042513808308354926824, 9.872569097821498153211605325271, 11.42060735138986702717471454937, 14.48108319706892555650423726979, 15.10406713732621319205090463113, 16.58214814139087710492790508983, 17.89394873153397213950548410865