L(s) = 1 | − i·5-s + 3.41i·7-s + 2.58·11-s + 3.41·13-s − 1.17i·17-s − 4.82·23-s − 25-s + 6i·29-s + 6.48i·31-s + 3.41·35-s + 9.07·37-s − 11.0i·41-s + 6.82i·43-s + 5.65·47-s − 4.65·49-s + ⋯ |
L(s) = 1 | − 0.447i·5-s + 1.29i·7-s + 0.779·11-s + 0.946·13-s − 0.284i·17-s − 1.00·23-s − 0.200·25-s + 1.11i·29-s + 1.16i·31-s + 0.577·35-s + 1.49·37-s − 1.72i·41-s + 1.04i·43-s + 0.825·47-s − 0.665·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.873473361\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.873473361\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 3.41iT - 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 6.48iT - 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 11.0iT - 41T^{2} \) |
| 43 | \( 1 - 6.82iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−8.970492416499444429082742592648, −8.345705574839226253370590200004, −7.44375135434468858331757518154, −6.40041371575159923065587357241, −5.88189923629957864994329291679, −5.10578831800880141340451404017, −4.16102517966306178938739157004, −3.25947593760492788234035029807, −2.18620615547437263837927221967, −1.16545672682995883583708282154,
0.68403806485071342146713917172, 1.81147926218948905926383692612, 3.09562898941084422017937544757, 4.14410115823240764998891539091, 4.26767265780100038559514460739, 5.98882469621893104502891328078, 6.20522813531020185721345245495, 7.24978720190498031848212027058, 7.78431302477789346193336564779, 8.573218680345838243915655257234