| L(s) = 1 | + (−1.27 + 1.75i)2-s + (−0.347 − 1.06i)3-s + (−0.833 − 2.56i)4-s + (−2.16 − 0.546i)5-s + (2.31 + 0.752i)6-s − 2.57·7-s + (1.43 + 0.467i)8-s + (1.40 − 1.02i)9-s + (3.72 − 3.10i)10-s + (−1.51 + 2.09i)11-s + (−2.45 + 1.78i)12-s + (1.80 + 2.48i)13-s + (3.27 − 4.51i)14-s + (0.169 + 2.50i)15-s + (1.71 − 1.24i)16-s + (3.25 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.900 + 1.23i)2-s + (−0.200 − 0.617i)3-s + (−0.416 − 1.28i)4-s + (−0.969 − 0.244i)5-s + (0.946 + 0.307i)6-s − 0.972·7-s + (0.508 + 0.165i)8-s + (0.468 − 0.340i)9-s + (1.17 − 0.982i)10-s + (−0.457 + 0.630i)11-s + (−0.708 + 0.514i)12-s + (0.500 + 0.689i)13-s + (0.876 − 1.20i)14-s + (0.0436 + 0.647i)15-s + (0.428 − 0.311i)16-s + (0.788 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.459989 + 0.325378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459989 + 0.325378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.16 + 0.546i)T \) |
| 17 | \( 1 + (-3.25 - 2.53i)T \) |
| good | 2 | \( 1 + (1.27 - 1.75i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.347 + 1.06i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + (1.51 - 2.09i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 2.48i)T + (-4.01 + 12.3i)T^{2} \) |
| 19 | \( 1 + (-0.831 + 2.56i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.46 - 3.24i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-7.26 + 2.36i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.00 - 1.62i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.672 + 0.488i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.01 - 5.52i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 9.25iT - 43T^{2} \) |
| 47 | \( 1 + (8.37 - 2.72i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (12.7 - 4.15i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (4.88 - 3.54i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.99 + 9.62i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-13.1 - 4.26i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.636 + 0.206i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.34 - 4.61i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.25 - 1.38i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.25 - 2.03i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.87 - 7.17i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.29 + 7.07i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.37682751704433956592045525511, −10.01652958071483500064292557694, −9.384375394396791469036897432960, −8.342206014121612699115174558123, −7.59781953121161967307100436622, −6.80603915191224993953112176805, −6.28227285557880533578976402384, −4.82522364737644592464136569383, −3.37373723962281790902125725798, −0.932442563455169627308835370259,
0.73152981764092339597886490079, 2.96963403164157514251004386694, 3.46873134064049889411751313152, 4.87891907688236940157618175567, 6.36016755689051933227802973231, 7.75676077520936781632608003087, 8.431373450067326701445473631736, 9.583746281725813085812171384671, 10.22429327130500786039358902464, 10.83394453297520420139170822558
