L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s − 1.00i·6-s + i·7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + (−1 − i)13-s + (0.707 − 0.707i)14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (1 + i)19-s + (−0.707 + 0.707i)21-s + 1.00·22-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s − 1.00i·6-s + i·7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + (−1 − i)13-s + (0.707 − 0.707i)14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (1 + i)19-s + (−0.707 + 0.707i)21-s + 1.00·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7886570797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7886570797\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.104743692162840015991582229119, −8.286062609035717701092674136532, −7.70306881845802960930203541215, −7.26449505944484082453238310108, −5.61929077008526890949319899612, −5.15803561652222012755930574629, −4.10146448574086131006820655679, −3.19314376998177353432153886406, −2.56890252297312000046964673872, −1.79520814450877843079955475644,
0.49608840477102763492301179845, 1.71580070647071614491177051355, 2.68201806813300423558113843435, 3.79596689472966908656620266538, 4.83927425243526971299563440970, 5.64022633051779227425812840371, 6.69039857964553564530477812456, 7.21973278813186634488805335180, 7.53434386109938538704011423547, 8.394746797372499003900357274229