L(s) = 1 | − 2.13e6i·2-s + (−3.48e9 − 9.86e9i)3-s − 1.66e11·4-s + 7.27e14i·5-s + (−2.10e16 + 7.45e15i)6-s − 7.17e17·7-s − 9.04e18i·8-s + (−8.50e19 + 6.87e19i)9-s + 1.55e21·10-s + 5.19e21i·11-s + (5.80e20 + 1.64e21i)12-s + 2.30e23·13-s + 1.53e24i·14-s + (7.17e24 − 2.53e24i)15-s − 2.00e25·16-s − 3.57e25i·17-s + ⋯ |
L(s) = 1 | − 1.01i·2-s + (−0.333 − 0.942i)3-s − 0.0378·4-s + 1.52i·5-s + (−0.960 + 0.339i)6-s − 1.28·7-s − 0.980i·8-s + (−0.777 + 0.628i)9-s + 1.55·10-s + 0.701i·11-s + (0.0126 + 0.0356i)12-s + 0.931·13-s + 1.30i·14-s + (1.43 − 0.508i)15-s − 1.03·16-s − 0.517i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{43}{2})\) |
\(\approx\) |
\(1.657594474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.657594474\) |
\(L(22)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.48e9 + 9.86e9i)T \) |
good | 2 | \( 1 + 2.13e6iT - 4.39e12T^{2} \) |
| 5 | \( 1 - 7.27e14iT - 2.27e29T^{2} \) |
| 7 | \( 1 + 7.17e17T + 3.11e35T^{2} \) |
| 11 | \( 1 - 5.19e21iT - 5.47e43T^{2} \) |
| 13 | \( 1 - 2.30e23T + 6.10e46T^{2} \) |
| 17 | \( 1 + 3.57e25iT - 4.77e51T^{2} \) |
| 19 | \( 1 - 1.20e27T + 5.10e53T^{2} \) |
| 23 | \( 1 + 4.53e28iT - 1.55e57T^{2} \) |
| 29 | \( 1 - 3.09e30iT - 2.63e61T^{2} \) |
| 31 | \( 1 - 2.88e31T + 4.33e62T^{2} \) |
| 37 | \( 1 + 7.76e29T + 7.31e65T^{2} \) |
| 41 | \( 1 + 1.18e33iT - 5.45e67T^{2} \) |
| 43 | \( 1 - 1.46e33T + 4.03e68T^{2} \) |
| 47 | \( 1 - 4.50e34iT - 1.69e70T^{2} \) |
| 53 | \( 1 - 2.22e36iT - 2.62e72T^{2} \) |
| 59 | \( 1 + 3.90e36iT - 2.37e74T^{2} \) |
| 61 | \( 1 - 2.78e37T + 9.63e74T^{2} \) |
| 67 | \( 1 - 3.05e38T + 4.95e76T^{2} \) |
| 71 | \( 1 - 3.64e38iT - 5.66e77T^{2} \) |
| 73 | \( 1 - 1.42e39T + 1.81e78T^{2} \) |
| 79 | \( 1 - 3.60e38T + 5.01e79T^{2} \) |
| 83 | \( 1 - 7.96e39iT - 3.99e80T^{2} \) |
| 89 | \( 1 - 6.40e40iT - 7.48e81T^{2} \) |
| 97 | \( 1 - 5.27e41T + 2.78e83T^{2} \) |
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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
−15.90270843279288904424797914028, −13.79748237031635691693105750991, −12.38771310995479210120179746477, −11.14468723496899272152217619587, −9.938582683824295958594837622159, −7.12419506560109416011582736747, −6.36617728561207760595871763966, −3.26760080390058574678121160289, −2.49960711277215865982690254624, −0.856605047580397230726380938777,
0.75517479738336487920874185080, 3.50685068140775349797812985051, 5.23832066071894484927133740510, 6.16522051574486351720231153244, 8.372468175561931707494604161803, 9.584032090709616642664044894812, 11.66605714338208955505236056047, 13.52563558109262358324083498001, 15.80924944524887802300426424027, 16.10801875091766915576246407085