| L(s) = 1 | − 2.41·3-s + (−1.94 − 1.10i)5-s − i·7-s + 2.84·9-s + 2.55i·11-s − 5.47·13-s + (4.70 + 2.66i)15-s − 2.44i·17-s + 4.99i·19-s + 2.41i·21-s + 1.10i·23-s + (2.56 + 4.29i)25-s + 0.376·27-s + 1.96i·29-s − 2.33·31-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + (−0.869 − 0.493i)5-s − 0.377i·7-s + 0.948·9-s + 0.770i·11-s − 1.51·13-s + (1.21 + 0.688i)15-s − 0.593i·17-s + 1.14i·19-s + 0.527i·21-s + 0.229i·23-s + (0.513 + 0.858i)25-s + 0.0724·27-s + 0.364i·29-s − 0.419·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4240394013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4240394013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.94 + 1.10i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 2.55iT - 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 - 4.99iT - 19T^{2} \) |
| 23 | \( 1 - 1.10iT - 23T^{2} \) |
| 29 | \( 1 - 1.96iT - 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 - 5.99T + 43T^{2} \) |
| 47 | \( 1 - 8.05iT - 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 + 1.44iT - 59T^{2} \) |
| 61 | \( 1 + 8.71iT - 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 + 7.55iT - 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 8.26T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−9.038541315068134286686060122607, −7.88778332462915124680883642852, −7.32703597879411124285658365333, −6.71523318209027570230533817366, −5.59338208588278785848768050804, −4.92222644220375804478414527849, −4.43327977774442207358952977886, −3.30084685494175208957552268325, −1.71964136532950683484962399754, −0.35679454214067493036728122805,
0.54797253774785511162350369667, 2.35825661232715117792924858933, 3.41250344693788137205689979844, 4.53218878372597785729493302914, 5.18055123086473334910696603623, 5.96980018225278236067010085645, 6.83490373097605614659643289041, 7.29918618329988446485588619304, 8.335942750713447987001635158033, 9.073884136527103759516079389515
