L(s) = 1 | + 2-s + 1.41i·5-s + 7-s − 8-s + 1.41i·10-s + 1.41i·13-s + 14-s − 16-s − 17-s − 1.41i·19-s + 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯ |
L(s) = 1 | + 2-s + 1.41i·5-s + 7-s − 8-s + 1.41i·10-s + 1.41i·13-s + 14-s − 16-s − 17-s − 1.41i·19-s + 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.533509793\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.533509793\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 109 | \( 1 - T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.74663947989589951931268354125, −9.324345637051428307308548289705, −8.867304082089176527619433148591, −7.51047513547281795869911579275, −6.76086190507328051084022751702, −6.10426306347206423005389140422, −4.77122937907628909459488817351, −4.35796280912732380919952418850, −3.08468252826340446715338004736, −2.21892000096672711245653318261,
1.24759465372194726074540865491, 2.89495466716610130722555927148, 4.18552752286383439917806910603, 4.81745562616014935918832703365, 5.40934160941686382023532527703, 6.26149998094942425942717335482, 7.82217113020343557526561979391, 8.382074114355067426487825222951, 9.063264189166380558283060267607, 10.06298496255351722159907515631