| L(s) = 1 | + (0.921 + 1.07i)2-s + (−0.121 + 0.375i)3-s + (−0.303 + 1.97i)4-s + (0.445 − 2.19i)5-s + (−0.515 + 0.214i)6-s + (3.43 − 3.43i)7-s + (−2.40 + 1.49i)8-s + (2.30 + 1.67i)9-s + (2.76 − 1.54i)10-s + (0.833 + 0.132i)11-s + (−0.704 − 0.354i)12-s + (0.687 − 0.946i)13-s + (6.84 + 0.522i)14-s + (0.768 + 0.434i)15-s + (−3.81 − 1.19i)16-s + (−3.37 − 1.72i)17-s + ⋯ |
| L(s) = 1 | + (0.651 + 0.758i)2-s + (−0.0704 + 0.216i)3-s + (−0.151 + 0.988i)4-s + (0.199 − 0.979i)5-s + (−0.210 + 0.0876i)6-s + (1.29 − 1.29i)7-s + (−0.848 + 0.528i)8-s + (0.767 + 0.557i)9-s + (0.873 − 0.487i)10-s + (0.251 + 0.0398i)11-s + (−0.203 − 0.102i)12-s + (0.190 − 0.262i)13-s + (1.83 + 0.139i)14-s + (0.198 + 0.112i)15-s + (−0.954 − 0.299i)16-s + (−0.819 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98449 + 0.787217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98449 + 0.787217i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.921 - 1.07i)T \) |
| 5 | \( 1 + (-0.445 + 2.19i)T \) |
| good | 3 | \( 1 + (0.121 - 0.375i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.43 + 3.43i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.833 - 0.132i)T + (10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (-0.687 + 0.946i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.37 + 1.72i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (3.40 - 6.67i)T + (-11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-7.01 - 1.11i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.828 + 1.62i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (3.04 - 0.989i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.56 - 3.53i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.24 + 4.47i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 9.51iT - 43T^{2} \) |
| 47 | \( 1 + (4.56 - 2.32i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.04 + 6.30i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (2.54 - 0.403i)T + (56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (13.6 + 2.16i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-13.6 + 4.43i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.89 + 5.82i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.795 - 5.02i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.07 - 6.38i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.0485 - 0.149i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.947 + 0.688i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.12 + 10.0i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.38469897585387049044159059643, −10.64136976240915290320750181562, −9.404852664279662644900582720291, −8.258878027668688918989303655509, −7.72592161399080572427807266502, −6.70054343498348894873784594326, −5.28115550153996872105700265102, −4.60747201174469956442753785071, −3.92253024601913466074534632607, −1.59583826105232493899930423938,
1.75545680194399278836019300375, 2.69194998097279675483510719417, 4.18062782023992165295884715287, 5.20618110660669339071617194201, 6.32616799207018381263785870205, 7.08778170324548962155744472382, 8.779263905731621286101541249670, 9.316402633595713162074778543050, 10.82562956114671118094311236834, 11.07709249240186455189256902870
