L(s) = 1 | + (−1 + i)2-s − 2i·4-s − 2i·5-s − 7-s + (2 + 2i)8-s + (2 + 2i)10-s − 5i·13-s + (1 − i)14-s − 4·16-s + 17-s − 4i·19-s − 4·20-s − 5·23-s + 25-s + (5 + 5i)26-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − i·4-s − 0.894i·5-s − 0.377·7-s + (0.707 + 0.707i)8-s + (0.632 + 0.632i)10-s − 1.38i·13-s + (0.267 − 0.267i)14-s − 16-s + 0.242·17-s − 0.917i·19-s − 0.894·20-s − 1.04·23-s + 0.200·25-s + (0.980 + 0.980i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4606952248\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4606952248\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 13T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.935168772701952214420735604333, −8.547536848463692056793550593836, −7.57497846125810977950257510947, −6.96582062291966178022602929668, −5.74872591673051753796580018429, −5.37029345297608884360089259703, −4.32943916376496872714672702190, −2.93910021077498731887237177472, −1.43751376113978214378912348920, −0.23429795009191015451726674852,
1.67577352415054073113134653995, 2.60911414813695091113651228143, 3.65446722905062973768644923531, 4.33117788465248853942571840832, 5.93579477733754355163933939893, 6.71349829997619331367582564672, 7.46850419789980652659799474827, 8.239869611956688114807726572980, 9.192392812184509574549061191963, 9.858836300135979305976447501723