L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.923 − 0.382i)8-s + (−0.793 − 0.608i)9-s + (0.732 − 0.835i)11-s + (0.866 − 0.5i)16-s + (−0.866 − 0.499i)18-s + (0.617 − 0.923i)22-s + (−0.607 − 0.465i)23-s + (0.608 + 0.793i)25-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + (−0.923 − 0.382i)36-s + (−0.349 − 0.172i)37-s + (−0.382 + 1.92i)43-s + (0.491 − 0.996i)44-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.923 − 0.382i)8-s + (−0.793 − 0.608i)9-s + (0.732 − 0.835i)11-s + (0.866 − 0.5i)16-s + (−0.866 − 0.499i)18-s + (0.617 − 0.923i)22-s + (−0.607 − 0.465i)23-s + (0.608 + 0.793i)25-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + (−0.923 − 0.382i)36-s + (−0.349 − 0.172i)37-s + (−0.382 + 1.92i)43-s + (0.491 − 0.996i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.363366114\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.363366114\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 5 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 11 | \( 1 + (-0.732 + 0.835i)T + (-0.130 - 0.991i)T^{2} \) |
| 13 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 23 | \( 1 + (0.607 + 0.465i)T + (0.258 + 0.965i)T^{2} \) |
| 29 | \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.349 + 0.172i)T + (0.608 + 0.793i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.47 - 1.29i)T + (0.130 + 0.991i)T^{2} \) |
| 59 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 61 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 67 | \( 1 + (-0.369 - 0.125i)T + (0.793 + 0.608i)T^{2} \) |
| 71 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 79 | \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.765588604356895120868486082997, −7.959852025021587436029203750519, −7.05111080185964139177939321946, −6.16026550193083150925662801679, −5.92176993798569449439008987163, −4.88644771445066357609992525859, −3.97062559107245348970893459153, −3.30112185774177797688668054203, −2.46516438905647237942780163237, −1.12625727255143989993265237055,
1.70604216548204204389263551256, 2.53757386055215854156814404671, 3.55960218375766458078078367828, 4.31221291218420293808450014372, 5.21789687661479266776628487422, 5.70422577769745074426805261912, 6.82385756084322126505761836525, 7.10804676112820154487190359140, 8.205737209823415494728752507779, 8.718132311937121356806560123381