| L(s) = 1 | + (0.838 + 0.544i)2-s + (−0.309 + 2.34i)3-s + (0.406 + 0.913i)4-s + (−3.21 + 2.60i)5-s + (−1.53 + 1.80i)6-s + (2.64 − 0.110i)7-s + (−0.156 + 0.987i)8-s + (−2.52 − 0.677i)9-s + (−4.11 + 0.432i)10-s + (2.97 + 5.48i)11-s + (−2.27 + 0.673i)12-s + (−0.294 − 3.74i)13-s + (2.27 + 1.34i)14-s + (−5.12 − 8.35i)15-s + (−0.669 + 0.743i)16-s + (1.50 − 5.08i)17-s + ⋯ |
| L(s) = 1 | + (0.593 + 0.385i)2-s + (−0.178 + 1.35i)3-s + (0.203 + 0.456i)4-s + (−1.43 + 1.16i)5-s + (−0.628 + 0.735i)6-s + (0.999 − 0.0417i)7-s + (−0.0553 + 0.349i)8-s + (−0.842 − 0.225i)9-s + (−1.30 + 0.136i)10-s + (0.897 + 1.65i)11-s + (−0.655 + 0.194i)12-s + (−0.0817 − 1.03i)13-s + (0.608 + 0.360i)14-s + (−1.32 − 2.15i)15-s + (−0.167 + 0.185i)16-s + (0.365 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0501031 - 1.58644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0501031 - 1.58644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.838 - 0.544i)T \) |
| 7 | \( 1 + (-2.64 + 0.110i)T \) |
| 41 | \( 1 + (0.923 - 6.33i)T \) |
| good | 3 | \( 1 + (0.309 - 2.34i)T + (-2.89 - 0.776i)T^{2} \) |
| 5 | \( 1 + (3.21 - 2.60i)T + (1.03 - 4.89i)T^{2} \) |
| 11 | \( 1 + (-2.97 - 5.48i)T + (-5.99 + 9.22i)T^{2} \) |
| 13 | \( 1 + (0.294 + 3.74i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 5.08i)T + (-14.2 - 9.25i)T^{2} \) |
| 19 | \( 1 + (3.28 - 1.56i)T + (11.9 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.198 + 0.933i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-0.518 + 2.16i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (-0.189 - 1.79i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.197 + 1.88i)T + (-36.1 - 7.69i)T^{2} \) |
| 43 | \( 1 + (-4.13 + 2.10i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-3.92 - 0.728i)T + (43.8 + 16.8i)T^{2} \) |
| 53 | \( 1 + (1.20 - 1.27i)T + (-2.77 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-2.37 + 2.14i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.0460 + 0.879i)T + (-60.6 + 6.37i)T^{2} \) |
| 67 | \( 1 + (-0.401 - 15.3i)T + (-66.9 + 3.50i)T^{2} \) |
| 71 | \( 1 + (-13.5 - 8.28i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (1.97 + 7.36i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.69 + 5.13i)T + (20.4 + 76.3i)T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 + (2.17 - 6.13i)T + (-69.1 - 56.0i)T^{2} \) |
| 97 | \( 1 + (5.99 - 3.67i)T + (44.0 - 86.4i)T^{2} \) |
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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
−11.16634858785785281544530658927, −10.48006136845337120050591115231, −9.649603481324443334847327157593, −8.300358761041453112801894681862, −7.48855460045053498642668660647, −6.83129462176255881579843217699, −5.31354869741217896648076493221, −4.36047079191184113475914456184, −3.99825846887227775876887821706, −2.75135720375996989492584894781,
0.828894905385137920260243515966, 1.74334446149796625979214070994, 3.70765435129603832594867878272, 4.42959077494477914189846591688, 5.65684921353382175189792525913, 6.61532803776305014014209520252, 7.68132084129634046401144178878, 8.426649316330859072056444095927, 8.951498550995069893967245087415, 10.98424952560072369319714822120
