| 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 | \( 1 + (13 + 13i)T + 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66662421035005122413665592620, −9.813040406290643222927654857079, −8.892777616076392053113395387391, −7.86095739511536240670876666551, −7.05712736452429743387604982721, −6.06599255800950452550590219230, −5.37953566580412955310819941787, −3.85724090966648733626557431643, −2.98570042786697898155467027824, −1.58826915175872574550972198792,
0.935267726284866606307417625882, 2.38768127530993581969491773566, 3.80709863824822138295433709877, 4.93054464557888470245271706652, 5.66122178961127731544996289487, 6.69874058099085780973968331031, 7.82855208493149221467800522786, 8.538180891490929448922770681746, 9.629131391752696114180756406965, 9.898988187197046592924298763821
