| L(s) = 1 | + (0.535 − 0.844i)2-s + (1.01 − 0.955i)3-s + (−0.425 − 0.904i)4-s + (1.44 − 1.71i)5-s + (−0.261 − 1.37i)6-s + (−0.793 + 0.576i)7-s + (−0.992 − 0.125i)8-s + (−0.0661 + 1.05i)9-s + (−0.672 − 2.13i)10-s + (0.690 − 1.08i)11-s + (−1.29 − 0.513i)12-s + (0.0172 − 0.274i)13-s + (0.0615 + 0.978i)14-s + (−0.168 − 3.11i)15-s + (−0.637 + 0.770i)16-s + (−2.32 + 4.94i)17-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.597i)2-s + (0.587 − 0.551i)3-s + (−0.212 − 0.452i)4-s + (0.644 − 0.764i)5-s + (−0.106 − 0.559i)6-s + (−0.299 + 0.217i)7-s + (−0.350 − 0.0443i)8-s + (−0.0220 + 0.350i)9-s + (−0.212 − 0.674i)10-s + (0.208 − 0.327i)11-s + (−0.374 − 0.148i)12-s + (0.00479 − 0.0762i)13-s + (0.0164 + 0.261i)14-s + (−0.0436 − 0.804i)15-s + (−0.159 + 0.192i)16-s + (−0.564 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0134 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0134 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28293 - 1.30032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28293 - 1.30032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 5 | \( 1 + (-1.44 + 1.71i)T \) |
| good | 3 | \( 1 + (-1.01 + 0.955i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (0.793 - 0.576i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.690 + 1.08i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (-0.0172 + 0.274i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (2.32 - 4.94i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (0.872 + 0.819i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.03 - 0.264i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (-3.72 - 2.04i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (-2.54 + 5.40i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (4.07 - 4.92i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-0.531 + 0.136i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-0.891 + 2.74i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (5.83 - 0.737i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (0.525 - 2.75i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (-4.73 - 1.87i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-12.9 - 3.31i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (9.21 - 5.06i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (6.57 - 0.830i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-7.34 + 2.90i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-8.06 + 7.57i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (6.18 + 5.80i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (15.6 - 6.19i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (13.5 + 7.47i)T + (51.9 + 81.8i)T^{2} \) |
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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
−12.08764328777614464110691097799, −10.88556029282597371484761090176, −9.933466507152667535455960325654, −8.838436390152758106268768489946, −8.233211651891112213052018898264, −6.63545241567435397983246631538, −5.58294485505133016407951670820, −4.35349746295430270326188122693, −2.76668612934505843807759401178, −1.57081506168328572423499612346,
2.67595733198380510017634828803, 3.76595473616857657712806217936, 5.04957315423869106054126792873, 6.46638959133231123205163450137, 7.03480713254849966155448782343, 8.478652770388655887718566757604, 9.460192571027005281821069733450, 10.09460888140662824584203160537, 11.33431658740992310876484979763, 12.46813272009436294806783612461
