L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.608 − 0.793i)5-s + (0.707 + 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 − 0.608i)10-s + (0.541 − 0.541i)13-s + (0.500 + 0.866i)16-s + (−1.78 + 0.478i)17-s + (−0.965 + 0.258i)18-s + (0.923 − 0.382i)20-s + (−0.258 − 0.965i)25-s + (0.662 − 0.382i)26-s − 1.41i·29-s + (0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.608 − 0.793i)5-s + (0.707 + 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 − 0.608i)10-s + (0.541 − 0.541i)13-s + (0.500 + 0.866i)16-s + (−1.78 + 0.478i)17-s + (−0.965 + 0.258i)18-s + (0.923 − 0.382i)20-s + (−0.258 − 0.965i)25-s + (0.662 − 0.382i)26-s − 1.41i·29-s + (0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.854578233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.854578233\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (-0.608 + 0.793i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 17 | \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.84iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
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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
−10.46996942682809432529551970871, −9.252486139113314136788090848636, −8.396308862133121437684910332637, −7.82537808286554002364933869104, −6.35466400736084675887059108596, −6.00772387891417105260301006291, −4.95483051265858349701604548678, −4.28793508650897871333503151090, −2.90908789726601974994207346512, −1.90046720803874689887790378417,
1.91222798708511541510137391799, 2.87879930550531657527706266394, 3.78802929632220423635747397014, 4.95437057085247171188349313022, 5.91250629484575205709951252964, 6.60236622594144803465030695798, 7.18761584136018293092489817723, 8.718874969793980488200395216915, 9.387270734767369360800213809437, 10.50708976596333280299059387853