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
−9.479743610352975667477366710808, −8.478971645240419979061707877664, −7.67727900089923745901563411119, −6.92651663517791171334911762347, −6.45378846783430724973504876313, −5.17238506218695064463017763902, −4.41769376757482740190142269420, −3.33411535201120495182752487767, −2.57479472643802334869460541131, −1.10946425705734464540746263077,
0.30218725626771193379447580027, 1.80829386144890874222903171756, 2.70217613554113964398766267539, 3.89864238529919171312906771807, 5.06201422928925150182458618438, 5.40933485150925272789909067150, 6.41133472042718887265901975168, 7.43965784432181499084550704060, 8.220238591830687256829988682795, 9.069196781759332870426186988936