L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.806 + 2.48i)3-s + (0.309 − 0.951i)4-s + (1.11 + 1.93i)5-s + (0.806 + 2.48i)6-s − 1.68·7-s + (−0.309 − 0.951i)8-s + (−3.08 − 2.23i)9-s + (2.04 + 0.914i)10-s + (−4.29 + 3.12i)11-s + (2.11 + 1.53i)12-s + (−2.41 − 1.75i)13-s + (−1.36 + 0.993i)14-s + (−5.71 + 1.19i)15-s + (−0.809 − 0.587i)16-s + (−0.256 − 0.789i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.465 + 1.43i)3-s + (0.154 − 0.475i)4-s + (0.497 + 0.867i)5-s + (0.329 + 1.01i)6-s − 0.638·7-s + (−0.109 − 0.336i)8-s + (−1.02 − 0.746i)9-s + (0.645 + 0.289i)10-s + (−1.29 + 0.941i)11-s + (0.609 + 0.442i)12-s + (−0.669 − 0.486i)13-s + (−0.365 + 0.265i)14-s + (−1.47 + 0.309i)15-s + (−0.202 − 0.146i)16-s + (−0.0622 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0387138 + 0.869797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0387138 + 0.869797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-1.11 - 1.93i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (0.806 - 2.48i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 1.68T + 7T^{2} \) |
| 11 | \( 1 + (4.29 - 3.12i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.41 + 1.75i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.256 + 0.789i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + (0.415 - 0.302i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 5.77i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.993 - 3.05i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.19 - 3.77i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.06 + 0.776i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + (2.26 - 6.97i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.780 + 2.40i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.78 + 4.20i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.439 - 0.319i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.07 - 12.5i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.320 - 0.986i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.2 - 8.88i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.637 + 1.96i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.08 - 3.33i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.79 + 2.76i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.04 - 12.4i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.31074749606976443128733914262, −9.940632619558567501242506662832, −9.436175107896515014692589808665, −7.84148066090114214194347617661, −6.79411095645060740069437698849, −5.83561622836708060818495087052, −5.07663833585879220206361068877, −4.34116392580452031994671721528, −3.13400976472028759780624220591, −2.46858236724742862905097778214,
0.32478857226671338142847748307, 1.89923539512055959314924879513, 3.00362135223477423235652707343, 4.60359864902259567701008430062, 5.59199923192171537639065822833, 6.07630570701481208748996387094, 6.94726412287041173309009616643, 7.80801479642247971044618716570, 8.469421024854344683741940241479, 9.510364031557510046758738477422