L(s) = 1 | + (0.5 + 0.866i)5-s + (1.87 + 1.86i)7-s + (1.56 − 2.71i)11-s − 3.53·13-s + (−2.76 + 4.78i)17-s + (2.94 + 5.10i)19-s + (0.0732 + 0.126i)23-s + (−0.499 + 0.866i)25-s − 2.91·29-s + (−2.54 + 4.40i)31-s + (−0.678 + 2.55i)35-s + (−0.321 − 0.556i)37-s + 8.63·41-s − 5.93·43-s + (−0.704 − 1.22i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (0.708 + 0.705i)7-s + (0.471 − 0.817i)11-s − 0.981·13-s + (−0.669 + 1.16i)17-s + (0.676 + 1.17i)19-s + (0.0152 + 0.0264i)23-s + (−0.0999 + 0.173i)25-s − 0.542·29-s + (−0.457 + 0.791i)31-s + (−0.114 + 0.432i)35-s + (−0.0528 − 0.0914i)37-s + 1.34·41-s − 0.904·43-s + (−0.102 − 0.178i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.609348349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609348349\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.87 - 1.86i)T \) |
good | 11 | \( 1 + (-1.56 + 2.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 - 5.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0732 - 0.126i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.91T + 29T^{2} \) |
| 31 | \( 1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.321 + 0.556i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + (0.704 + 1.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.83 + 8.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.67 - 2.89i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.19 - 10.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.62 - 11.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.658T + 71T^{2} \) |
| 73 | \( 1 + (-2.34 + 4.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.67 + 4.63i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + (-7.20 - 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.86T + 97T^{2} \) |
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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
−8.989847285266329058789792544495, −8.441538025738924188322853492341, −7.65221999095256195011791301145, −6.82655696562595818377915314775, −5.85514624235916899103748365742, −5.45276214461923777039309541160, −4.31758083602915070710547707863, −3.42279734142020981033168943266, −2.35794738457325678386649084021, −1.46045844916021530562191624104,
0.52767848147610566907302002268, 1.80933821284886315757397538578, 2.73762236733699511394490400822, 4.10947606533237581650639494257, 4.76325298860100882342295641516, 5.28762584806046058061293727037, 6.56743516331238987480480126077, 7.32989615421576428887214128922, 7.66314009518039564397904575782, 8.871427720004670615607536509417