| L(s) = 1 | + (0.150 + 0.586i)2-s + (2.96 − 0.566i)3-s + (1.43 − 0.786i)4-s + (0.779 + 1.65i)6-s + (−2.97 + 4.09i)7-s + (1.50 + 1.60i)8-s + (5.69 − 2.25i)9-s + (0.644 − 0.165i)11-s + (3.80 − 3.14i)12-s + (−0.641 − 1.62i)13-s + (−2.85 − 1.12i)14-s + (1.03 − 1.63i)16-s + (0.790 − 1.43i)17-s + (2.18 + 3.00i)18-s + (−0.166 + 0.873i)19-s + ⋯ |
| L(s) = 1 | + (0.106 + 0.414i)2-s + (1.71 − 0.326i)3-s + (0.715 − 0.393i)4-s + (0.318 + 0.675i)6-s + (−1.12 + 1.54i)7-s + (0.532 + 0.567i)8-s + (1.89 − 0.752i)9-s + (0.194 − 0.0499i)11-s + (1.09 − 0.908i)12-s + (−0.177 − 0.449i)13-s + (−0.762 − 0.301i)14-s + (0.259 − 0.408i)16-s + (0.191 − 0.348i)17-s + (0.514 + 0.707i)18-s + (−0.0382 + 0.200i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.91079 + 0.568792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.91079 + 0.568792i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.150 - 0.586i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (-2.96 + 0.566i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (2.97 - 4.09i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.644 + 0.165i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (0.641 + 1.62i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-0.790 + 1.43i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (0.166 - 0.873i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (3.83 + 0.241i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (4.85 + 0.613i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (4.26 + 2.34i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (1.87 + 1.19i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.0553 - 0.879i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-5.21 + 1.69i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-4.01 + 4.27i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (5.99 + 2.82i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-2.85 - 3.45i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.497 - 7.90i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (0.307 + 2.43i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (-0.971 - 0.912i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (10.6 + 8.81i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-0.614 - 3.22i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-11.9 - 2.28i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (1.29 - 1.57i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (2.12 - 16.8i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.33556988186546505915366678335, −9.485830625529617022908375909716, −8.945931529966694689037868464975, −7.979991344515302428513815559625, −7.24176093108630579012095525699, −6.25281270030888136116786898754, −5.46561441075029490479710278968, −3.65243850949560946075766206838, −2.68742234030122734856891410353, −2.01066395091917420183066001924,
1.70276325925209667285881472417, 2.94624076490421436858567110739, 3.71218535122658835412471321812, 4.22693905355732010146772171283, 6.42207704533198558704777017517, 7.32931487576007217499844948043, 7.72841168229198124570799696754, 8.961796839871417507999369164857, 9.778176003970508143668794708544, 10.33749564983278477878443181697
