L(s) = 1 | + (−0.898 + 0.898i)3-s + (0.786 + 0.786i)5-s − i·7-s + 1.38i·9-s + (−2.15 − 2.15i)11-s + (−1.31 + 1.31i)13-s − 1.41·15-s − 1.79·17-s + (−0.531 + 0.531i)19-s + (0.898 + 0.898i)21-s − 5.95i·23-s − 3.76i·25-s + (−3.94 − 3.94i)27-s + (−5.62 + 5.62i)29-s + 4.34·31-s + ⋯ |
L(s) = 1 | + (−0.519 + 0.519i)3-s + (0.351 + 0.351i)5-s − 0.377i·7-s + 0.461i·9-s + (−0.650 − 0.650i)11-s + (−0.364 + 0.364i)13-s − 0.365·15-s − 0.436·17-s + (−0.121 + 0.121i)19-s + (0.196 + 0.196i)21-s − 1.24i·23-s − 0.752i·25-s + (−0.758 − 0.758i)27-s + (−1.04 + 1.04i)29-s + 0.779·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3727384071\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3727384071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.898 - 0.898i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.786 - 0.786i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.15 + 2.15i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.31 - 1.31i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + (0.531 - 0.531i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.95iT - 23T^{2} \) |
| 29 | \( 1 + (5.62 - 5.62i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + (2.12 + 2.12i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.712iT - 41T^{2} \) |
| 43 | \( 1 + (2.96 + 2.96i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + (-3.38 - 3.38i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.41 + 2.41i)T + 59iT^{2} \) |
| 61 | \( 1 + (7.09 - 7.09i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.76 + 6.76i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.96iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + (5.95 - 5.95i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−9.062798305674141892907435190337, −8.288757752741275147154733407852, −7.41050754264797852583179358971, −6.54182984991062586527507018165, −5.76887969467790972932579228082, −4.91142449790652749729234603515, −4.22479237345333491891963807425, −2.99266534025468567870587983937, −1.99939378087703272285589497352, −0.14456743821284355324239685537,
1.38109620926212028316898158016, 2.42972181836609897740625077363, 3.64042876588065855614341601996, 4.87102938378772231999959597886, 5.53000131422166637609800240030, 6.25560960313831640848957936214, 7.17011941094087745184036480675, 7.79715257282842307156794926418, 8.802025205847102799910885100725, 9.607159697436320053483469599838