L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s − 3.12·7-s + 8-s − 9-s − 5.12i·11-s + i·12-s + (−0.561 + 3.56i)13-s − 3.12·14-s + 16-s − 2i·17-s − 18-s − 6i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s − 1.18·7-s + 0.353·8-s − 0.333·9-s − 1.54i·11-s + 0.288i·12-s + (−0.155 + 0.987i)13-s − 0.834·14-s + 0.250·16-s − 0.485i·17-s − 0.235·18-s − 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.697586244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.697586244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 5.12iT - 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.12iT - 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 0.876iT - 41T^{2} \) |
| 43 | \( 1 - 6.24iT - 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 1.12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 3.12iT - 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.128875823739147959132371387978, −8.412213934045844888323359274706, −7.18319925876117602376224519769, −6.45098917695169475614582808379, −5.88632141225567000607664020857, −4.86838575473212484425680611547, −4.06207269127067581773464405488, −3.17699410369377089317497460691, −2.51111692316063517798488232145, −0.46751513809357784199950439119,
1.47366500635121843456503006202, 2.58074905927218240663563321637, 3.49133835812813009246597914448, 4.34080604422450643156008685930, 5.61017041295661513107599635117, 5.96691338211739512667102550059, 7.11380313485256293861279780746, 7.40232068643149129623444784956, 8.390121944961545968247982894397, 9.559169485536193761824561149024