| L(s) = 1 | + (0.201 + 0.784i)2-s + (−2.56 + 0.488i)3-s + (1.17 − 0.647i)4-s + (−0.899 − 1.91i)6-s + (0.380 − 0.523i)7-s + (1.85 + 1.97i)8-s + (3.54 − 1.40i)9-s + (−0.370 + 0.0950i)11-s + (−2.70 + 2.23i)12-s + (−2.38 − 6.01i)13-s + (0.486 + 0.192i)14-s + (0.266 − 0.420i)16-s + (−1.75 + 3.20i)17-s + (1.81 + 2.49i)18-s + (1.14 − 5.99i)19-s + ⋯ |
| L(s) = 1 | + (0.142 + 0.554i)2-s + (−1.47 + 0.282i)3-s + (0.589 − 0.323i)4-s + (−0.367 − 0.780i)6-s + (0.143 − 0.197i)7-s + (0.655 + 0.697i)8-s + (1.18 − 0.467i)9-s + (−0.111 + 0.0286i)11-s + (−0.780 + 0.645i)12-s + (−0.660 − 1.66i)13-s + (0.130 + 0.0515i)14-s + (0.0667 − 0.105i)16-s + (−0.426 + 0.776i)17-s + (0.427 + 0.587i)18-s + (0.262 − 1.37i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11862 - 0.0767791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11862 - 0.0767791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.201 - 0.784i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (2.56 - 0.488i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-0.380 + 0.523i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (0.370 - 0.0950i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (2.38 + 6.01i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (1.75 - 3.20i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-1.14 + 5.99i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-6.01 - 0.378i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-3.96 - 0.501i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-0.629 - 0.346i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (2.57 + 1.63i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.0888 + 1.41i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-9.37 + 3.04i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.91 + 9.49i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (2.37 + 1.11i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (1.48 + 1.80i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.689 - 10.9i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (0.0669 + 0.529i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (5.36 + 5.03i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-2.27 - 1.88i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (0.744 + 3.90i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (8.40 + 1.60i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-5.03 + 6.08i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (0.470 - 3.72i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.65181346837179908832542193526, −10.23049242596825207044621573574, −8.795754278625776608644312189396, −7.50283818250706858708184739100, −6.92568718729601017500084689819, −5.89518893542518820945381731832, −5.30846963577641878647412153386, −4.57160657934301575295204939289, −2.72333650772116095524285715019, −0.792432768822148228475218187413,
1.31664669972365553715972665909, 2.60224298253859258576859590519, 4.18729642557234225494440719283, 5.06105062652297571762493838361, 6.23301936796994782240070260824, 6.89094423490275341151592125067, 7.64309049632124163600032648175, 9.109071238103948375505046802687, 10.10242490360033743768253394550, 11.01493889683074749505302154222
