L(s) = 1 | − 3.99i·5-s + 7.74i·7-s − 7.68·11-s + 1.42i·13-s − 22.3·17-s + 21.9·19-s + 18.1i·23-s + 9.02·25-s − 15.2i·29-s − 10.2i·31-s + 30.9·35-s − 59.1i·37-s − 41.3·41-s + 11.0·43-s − 63.6i·47-s + ⋯ |
L(s) = 1 | − 0.799i·5-s + 1.10i·7-s − 0.699·11-s + 0.109i·13-s − 1.31·17-s + 1.15·19-s + 0.789i·23-s + 0.360·25-s − 0.527i·29-s − 0.329i·31-s + 0.884·35-s − 1.59i·37-s − 1.00·41-s + 0.255·43-s − 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.065822135\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.065822135\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.99iT - 25T^{2} \) |
| 7 | \( 1 - 7.74iT - 49T^{2} \) |
| 11 | \( 1 + 7.68T + 121T^{2} \) |
| 13 | \( 1 - 1.42iT - 169T^{2} \) |
| 17 | \( 1 + 22.3T + 289T^{2} \) |
| 19 | \( 1 - 21.9T + 361T^{2} \) |
| 23 | \( 1 - 18.1iT - 529T^{2} \) |
| 29 | \( 1 + 15.2iT - 841T^{2} \) |
| 31 | \( 1 + 10.2iT - 961T^{2} \) |
| 37 | \( 1 + 59.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 11.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 164.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 35.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 147.T + 9.40e3T^{2} \) |
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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
−9.069196781759332870426186988936, −8.220238591830687256829988682795, −7.43965784432181499084550704060, −6.41133472042718887265901975168, −5.40933485150925272789909067150, −5.06201422928925150182458618438, −3.89864238529919171312906771807, −2.70217613554113964398766267539, −1.80829386144890874222903171756, −0.30218725626771193379447580027,
1.10946425705734464540746263077, 2.57479472643802334869460541131, 3.33411535201120495182752487767, 4.41769376757482740190142269420, 5.17238506218695064463017763902, 6.45378846783430724973504876313, 6.92651663517791171334911762347, 7.67727900089923745901563411119, 8.478971645240419979061707877664, 9.479743610352975667477366710808