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
−9.858836300135979305976447501723, −9.192392812184509574549061191963, −8.239869611956688114807726572980, −7.46850419789980652659799474827, −6.71349829997619331367582564672, −5.93579477733754355163933939893, −4.33117788465248853942571840832, −3.65446722905062973768644923531, −2.60911414813695091113651228143, −1.67577352415054073113134653995,
0.23429795009191015451726674852, 1.43751376113978214378912348920, 2.93910021077498731887237177472, 4.32943916376496872714672702190, 5.37029345297608884360089259703, 5.74872591673051753796580018429, 6.96582062291966178022602929668, 7.57497846125810977950257510947, 8.547536848463692056793550593836, 8.935168772701952214420735604333