L(s) = 1 | − i·3-s + 1.41i·5-s − 4.24·7-s − 9-s + i·11-s − 1.41i·13-s + 1.41·15-s + 4·17-s + 4.24i·21-s − 1.41·23-s + 2.99·25-s + i·27-s − 2.82i·29-s − 5.65·31-s + 33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.632i·5-s − 1.60·7-s − 0.333·9-s + 0.301i·11-s − 0.392i·13-s + 0.365·15-s + 0.970·17-s + 0.925i·21-s − 0.294·23-s + 0.599·25-s + 0.192i·27-s − 0.525i·29-s − 1.01·31-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.238184958\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238184958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−9.202287276405976489809784913535, −8.002159410045603329273532934357, −7.40078821027832404720984186892, −6.58386866247054288494199204581, −6.12193203087345977418770606191, −5.19372378781169219841924412317, −3.74376599208466115911212326859, −3.13870069576278663626368735108, −2.21817711603743526920600362874, −0.59554440926842071164938189252,
0.867723667013143349291215985481, 2.57113746929389983522732022347, 3.50989818419917860475227127120, 4.15018762383855522994068924009, 5.33735300701644208319057189307, 5.88257873752968549928619662561, 6.81617496001461916901756040339, 7.61359299113341571109928140303, 8.832230891460796331636933449199, 9.079548974482941301096031426969