L(s) = 1 | + (1 − 2i)5-s + (1 + i)13-s + (5 − 5i)17-s + (−3 − 4i)25-s − 10i·29-s + (−7 + 7i)37-s + 10·41-s − 7i·49-s + (−5 − 5i)53-s + 12·61-s + (3 − i)65-s + (11 + 11i)73-s + (−5 − 15i)85-s + 10i·89-s + (−13 + 13i)97-s + ⋯ |
L(s) = 1 | + (0.447 − 0.894i)5-s + (0.277 + 0.277i)13-s + (1.21 − 1.21i)17-s + (−0.600 − 0.800i)25-s − 1.85i·29-s + (−1.15 + 1.15i)37-s + 1.56·41-s − i·49-s + (−0.686 − 0.686i)53-s + 1.53·61-s + (0.372 − 0.124i)65-s + (1.28 + 1.28i)73-s + (−0.542 − 1.62i)85-s + 1.05i·89-s + (−1.31 + 1.31i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42303 - 0.793392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42303 - 0.793392i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5 + 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11 - 11i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + (13 - 13i)T - 97iT^{2} \) |
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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
−9.898988187197046592924298763821, −9.629131391752696114180756406965, −8.538180891490929448922770681746, −7.82855208493149221467800522786, −6.69874058099085780973968331031, −5.66122178961127731544996289487, −4.93054464557888470245271706652, −3.80709863824822138295433709877, −2.38768127530993581969491773566, −0.935267726284866606307417625882,
1.58826915175872574550972198792, 2.98570042786697898155467027824, 3.85724090966648733626557431643, 5.37953566580412955310819941787, 6.06599255800950452550590219230, 7.05712736452429743387604982721, 7.86095739511536240670876666551, 8.892777616076392053113395387391, 9.813040406290643222927654857079, 10.66662421035005122413665592620