L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−1.36 + 0.366i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.999 + i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (−1.36 − 0.366i)21-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−1.36 + 0.366i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.999 + i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (−1.36 − 0.366i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.858941924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.858941924\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.909548938459224123826373486926, −8.150784001851725724846763338634, −6.98588224065946504294983450466, −6.37223620684964732547095212043, −5.48408188285151495736270382229, −4.52389485981336885695173449877, −3.71675972076837110817201387967, −3.22625421081321184733950654863, −2.59031578627343758143073763606, −0.75817892646865677465865449856,
2.00249803956130361674357637052, 2.93241339511422814208722120099, 3.58760827757220044749053971400, 4.23612689281309335451332676922, 5.25229012724900773029402960015, 6.51510787029417149746854569528, 6.95546480873060149046200675098, 7.40031510319039315161524298805, 8.021162907752934314809722603729, 9.193654007379919206475678910586