| 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
−9.510364031557510046758738477422, −8.469421024854344683741940241479, −7.80801479642247971044618716570, −6.94726412287041173309009616643, −6.07630570701481208748996387094, −5.59199923192171537639065822833, −4.60359864902259567701008430062, −3.00362135223477423235652707343, −1.89923539512055959314924879513, −0.32478857226671338142847748307,
2.46858236724742862905097778214, 3.13400976472028759780624220591, 4.34116392580452031994671721528, 5.07663833585879220206361068877, 5.83561622836708060818495087052, 6.79411095645060740069437698849, 7.84148066090114214194347617661, 9.436175107896515014692589808665, 9.940632619558567501242506662832, 10.31074749606976443128733914262
