L(s) = 1 | − 2.33i·2-s + (−1.41 + 1.00i)3-s − 3.44·4-s + 4.22·5-s + (2.33 + 3.29i)6-s + (2.13 + 1.56i)7-s + 3.36i·8-s + (0.990 − 2.83i)9-s − 9.84i·10-s + i·11-s + (4.86 − 3.45i)12-s − 1.39i·13-s + (3.65 − 4.97i)14-s + (−5.96 + 4.23i)15-s + 0.965·16-s − 1.76·17-s + ⋯ |
L(s) = 1 | − 1.64i·2-s + (−0.815 + 0.578i)3-s − 1.72·4-s + 1.88·5-s + (0.954 + 1.34i)6-s + (0.805 + 0.592i)7-s + 1.18i·8-s + (0.330 − 0.943i)9-s − 3.11i·10-s + 0.301i·11-s + (1.40 − 0.996i)12-s − 0.387i·13-s + (0.977 − 1.32i)14-s + (−1.53 + 1.09i)15-s + 0.241·16-s − 0.427·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0168 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0168 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.878417 - 0.893344i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.878417 - 0.893344i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.41 - 1.00i)T \) |
| 7 | \( 1 + (-2.13 - 1.56i)T \) |
| 11 | \( 1 - iT \) |
good | 2 | \( 1 + 2.33iT - 2T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 13 | \( 1 + 1.39iT - 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 - 0.456iT - 19T^{2} \) |
| 23 | \( 1 + 3.57iT - 23T^{2} \) |
| 29 | \( 1 + 6.15iT - 29T^{2} \) |
| 31 | \( 1 - 2.68iT - 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 + 0.712T + 47T^{2} \) |
| 53 | \( 1 - 10.2iT - 53T^{2} \) |
| 59 | \( 1 + 9.93T + 59T^{2} \) |
| 61 | \( 1 + 0.721iT - 61T^{2} \) |
| 67 | \( 1 - 2.01T + 67T^{2} \) |
| 71 | \( 1 - 15.3iT - 71T^{2} \) |
| 73 | \( 1 + 7.85iT - 73T^{2} \) |
| 79 | \( 1 + 7.33T + 79T^{2} \) |
| 83 | \( 1 - 9.00T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 6.09iT - 97T^{2} \) |
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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
−11.82789340003319928230778542752, −10.85982175291863016153849767910, −10.23551459478219160582561470807, −9.550609005200963854550073550646, −8.723869178967270265745013211988, −6.41957552583995412941521407264, −5.37125888923450866060635037285, −4.52632626711962274113040862398, −2.68708497512944565951340268433, −1.53237342359211497304323976903,
1.71583806621070718840810063512, 4.84011784639931363792299780219, 5.47057025687191021545664111618, 6.42147094634087666409084303462, 7.03729675175596256428325201440, 8.217419882862557734513295957100, 9.301871102735505648376990669686, 10.39187617581416188850964363779, 11.40612569552622131507565816169, 12.99633069667602358654608545157