| L(s) = 1 | + i·2-s + (0.420 − 1.68i)3-s − 4-s + (1.68 + 0.420i)6-s + (−2.57 − 0.595i)7-s − i·8-s + (−2.64 − 1.41i)9-s − 2.82i·11-s + (−0.420 + 1.68i)12-s + 3.36i·13-s + (0.595 − 2.57i)14-s + 16-s − 4.75·17-s + (1.41 − 2.64i)18-s + 5.59i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.242 − 0.970i)3-s − 0.5·4-s + (0.685 + 0.171i)6-s + (−0.974 − 0.224i)7-s − 0.353i·8-s + (−0.881 − 0.471i)9-s − 0.852i·11-s + (−0.121 + 0.485i)12-s + 0.931i·13-s + (0.159 − 0.688i)14-s + 0.250·16-s − 1.15·17-s + (0.333 − 0.623i)18-s + 1.28i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2881931153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2881931153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.420 + 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.57 + 0.595i)T \) |
| good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.36iT - 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 7.29iT - 23T^{2} \) |
| 29 | \( 1 + 0.500iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 - 3.32T + 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 53 | \( 1 + 8.58iT - 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 9.81iT - 71T^{2} \) |
| 73 | \( 1 - 5.53iT - 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−9.982052895491697338846458611014, −9.205999266596071643332555769232, −8.501347577333947676234974183557, −7.68180477172143826091209407521, −6.78285801053523556425237807372, −6.32072382597586729472157807952, −5.47366226820978704073913360593, −3.98728849423320946683734976719, −3.11756067549309444879621372579, −1.59498779195073362440153173897,
0.12073940989756411574864509315, 2.43021840467089211455405742856, 3.02387974098355034823967867019, 4.25247524401359809376519776452, 4.84354539778485550546653002319, 6.00419470540924499041358537480, 7.01088033898196769744460830822, 8.303439064531408014098184328447, 8.992196948845019940026617766315, 9.689257317898064328740148346326
