L(s) = 1 | + i·2-s − 4-s + (0.707 + 0.707i)5-s + 1.41·7-s − i·8-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + 1.41i·14-s + 16-s − 17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s + (0.707 − 0.707i)22-s + i·23-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (0.707 + 0.707i)5-s + 1.41·7-s − i·8-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + 1.41i·14-s + 16-s − 17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s + (0.707 − 0.707i)22-s + i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.292145568\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292145568\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−9.442239537999100904886356760569, −8.500218943081132870087816643589, −7.87386351484362660668228086988, −7.23872271225403484937018266483, −6.37868536208954240287024296718, −5.55978910583392285385163396250, −4.99282014844883649316989246006, −4.05802977341906386499302823398, −2.78365700511872524726253274281, −1.57335299944060337354564257844,
1.01710712023317427930031504358, 2.16327285279202566813070696185, 2.75183510207255080940771329695, 4.40886999868329333792093800180, 4.99390085124004374958723872221, 5.21723281758893574486414839805, 6.68243655666069059793955408899, 7.88659554238131427615156143458, 8.323644875006776803554879314842, 9.154680313907273444776197842452