| L(s) = 1 | + (1.80 − 1.31i)2-s + (−0.436 + 1.34i)3-s + (0.923 − 2.84i)4-s + (2.23 + 0.105i)5-s + (0.974 + 2.99i)6-s − 2.63·7-s + (−0.682 − 2.09i)8-s + (0.815 + 0.592i)9-s + (4.17 − 2.74i)10-s + (3.81 − 2.77i)11-s + (3.41 + 2.47i)12-s + (0.242 + 0.175i)13-s + (−4.76 + 3.46i)14-s + (−1.11 + 2.95i)15-s + (0.846 + 0.614i)16-s + (0.309 + 0.951i)17-s + ⋯ |
| L(s) = 1 | + (1.27 − 0.928i)2-s + (−0.251 + 0.774i)3-s + (0.461 − 1.42i)4-s + (0.998 + 0.0469i)5-s + (0.397 + 1.22i)6-s − 0.996·7-s + (−0.241 − 0.742i)8-s + (0.271 + 0.197i)9-s + (1.31 − 0.867i)10-s + (1.15 − 0.836i)11-s + (0.985 + 0.715i)12-s + (0.0671 + 0.0487i)13-s + (−1.27 + 0.925i)14-s + (−0.287 + 0.762i)15-s + (0.211 + 0.153i)16-s + (0.0749 + 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65562 - 0.816603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65562 - 0.816603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 - 0.105i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| good | 2 | \( 1 + (-1.80 + 1.31i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.436 - 1.34i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 + (-3.81 + 2.77i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.242 - 0.175i)T + (4.01 + 12.3i)T^{2} \) |
| 19 | \( 1 + (2.09 + 6.45i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (6.43 - 4.67i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.758 - 2.33i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.525 - 1.61i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.77 + 4.19i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.57 + 4.77i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 + (-2.11 + 6.51i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.29 - 10.1i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.67 + 2.67i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.59 - 1.88i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.260 + 0.802i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (3.47 - 10.6i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.6 + 9.16i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.83 - 8.71i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.322 + 0.992i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.17 + 3.03i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.75 - 8.48i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.07468778433100129216900934442, −10.43458981456640765421496542842, −9.651669227290742934461293634063, −8.858957651241339297888130375426, −6.81365650433881829237108333442, −5.95736842410528086770241921881, −5.15155829208964071543910035477, −4.00958958294340458778354298418, −3.23900788351891077658917380361, −1.81605785507126932817610876668,
1.79355985158301060177258648719, 3.53737514492085998324362126886, 4.57495174947693437786251444186, 5.98851797280346416338334724616, 6.37443042747818037738278998175, 6.92672942344811861328202026198, 8.128502770126545354130736657898, 9.649207984533841788498892289835, 10.08579731875624940888851915341, 12.01655407624029257507271586142
