L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001938205\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001938205\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.700112880000607070960059493996, −9.246254029239740980119232918786, −8.961563577808475987237338630687, −7.955594975165847919820021026106, −6.82588037278540464491630510377, −6.08151520254665653405581156349, −4.97588879065555881578862278760, −4.56925172812211455262890776688, −3.13573954958441692168331243465, −1.56960827996413838761891495272,
1.21075637368525368329866818591, 2.35238718539025944563366193926, 3.26204845692091433435079383733, 4.06596236240117556461458709393, 5.71640639534600054393975751587, 6.87715081433727202897705999294, 7.29191951928437660531015918005, 8.223806072998561826942099750806, 8.986290818922945568309886387979, 10.03449868006302145032090143066