L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.965 + 0.258i)3-s + (−0.499 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.258 − 0.965i)6-s + (−0.866 − 0.5i)7-s + 0.999·8-s + (−0.258 + 0.965i)10-s + (−0.707 + 0.707i)12-s + (−0.258 − 0.965i)13-s + 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)20-s + (−0.707 − 0.707i)21-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.965 + 0.258i)3-s + (−0.499 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.258 − 0.965i)6-s + (−0.866 − 0.5i)7-s + 0.999·8-s + (−0.258 + 0.965i)10-s + (−0.707 + 0.707i)12-s + (−0.258 − 0.965i)13-s + 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)20-s + (−0.707 − 0.707i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6231409370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6231409370\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (0.258 + 0.965i)T \) |
good | 3 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.492527561368671968172858746919, −8.653806113327553358064823870346, −7.939740662128963866244821651056, −7.53921051118460846435601477860, −6.09750792387131589746551924661, −4.73644775602777376701980751237, −3.80841428105035049665092492889, −3.32938365779503436559575671916, −2.26678334495370230901696414870, −0.51191966608628537758498591436,
1.99917101183202042559393808497, 3.09248506125046979356690517063, 4.02712648392300483031749574975, 5.26339975756036807468438310409, 6.37159764073773852814817909220, 6.94775405892864555611710230710, 7.65351297228481624734575856588, 8.473255227404949020900930978297, 9.000613327450645978647665340544, 9.765117842677891122186705367579