L(s) = 1 | + (0.604 − 1.27i)2-s + (−1.26 − 1.54i)4-s + (−0.217 − 0.217i)5-s − 3.81·7-s + (−2.74 + 0.688i)8-s + (−0.409 + 0.146i)10-s + (−3.83 + 3.83i)11-s + (−2.88 − 2.88i)13-s + (−2.30 + 4.87i)14-s + (−0.778 + 3.92i)16-s − 4.71i·17-s + (4.77 − 4.77i)19-s + (−0.0601 + 0.611i)20-s + (2.58 + 7.21i)22-s + 6.18i·23-s + ⋯ |
L(s) = 1 | + (0.427 − 0.904i)2-s + (−0.634 − 0.772i)4-s + (−0.0972 − 0.0972i)5-s − 1.44·7-s + (−0.969 + 0.243i)8-s + (−0.129 + 0.0463i)10-s + (−1.15 + 1.15i)11-s + (−0.801 − 0.801i)13-s + (−0.616 + 1.30i)14-s + (−0.194 + 0.980i)16-s − 1.14i·17-s + (1.09 − 1.09i)19-s + (−0.0134 + 0.136i)20-s + (0.550 + 1.53i)22-s + 1.28i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.136188 + 0.449394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.136188 + 0.449394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.604 + 1.27i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.217 + 0.217i)T + 5iT^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + (3.83 - 3.83i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.88 + 2.88i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.71iT - 17T^{2} \) |
| 19 | \( 1 + (-4.77 + 4.77i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.18iT - 23T^{2} \) |
| 29 | \( 1 + (0.322 - 0.322i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.20iT - 31T^{2} \) |
| 37 | \( 1 + (2.92 - 2.92i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 43 | \( 1 + (-0.505 - 0.505i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + (1.85 + 1.85i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.77 + 4.77i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.832 + 0.832i)T + 61iT^{2} \) |
| 67 | \( 1 + (6.48 - 6.48i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.47iT - 71T^{2} \) |
| 73 | \( 1 + 6.68iT - 73T^{2} \) |
| 79 | \( 1 + 9.28iT - 79T^{2} \) |
| 83 | \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 2.81T + 97T^{2} \) |
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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
−10.51238576167286123426764214839, −9.656215464339236471657003666813, −9.495512752586900201950644485527, −7.78534874363825968533460014271, −6.84638469196466235822128545158, −5.45386277255987336342731585723, −4.76942604805982777554539490271, −3.23863066451913807016588273396, −2.52408503803596404480265773592, −0.24246941451442454541968315031,
2.95030551382139276951137598015, 3.77384071416161971863188343377, 5.23820932724897901511496744349, 6.07882360479960539332514399008, 6.91417837085923445661724322885, 7.897740680274141988155794926272, 8.806295978537168032163052350946, 9.765731551696354540775315166965, 10.65147683669387710176140918382, 12.04141288867496538382525951409