| L(s) = 1 | − 4-s + 9-s + 16-s + 2·19-s − 2·29-s − 36-s + 49-s − 2·59-s − 64-s − 2·76-s − 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 2·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4-s + 9-s + 16-s + 2·19-s − 2·29-s − 36-s + 49-s − 2·59-s − 64-s − 2·76-s − 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 2·171-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.196912916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196912916\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.979156002578638622016660934646, −8.539400458459226246713354083138, −8.062701672845131715446036238713, −7.65045826460458957704852268444, −7.50949738233287366161816190885, −7.16898432497399265152039396491, −6.77686020155119560370831250674, −6.11287410826067766002001180262, −5.87099393414101980963581132704, −5.32327541962982309488265821620, −5.22767463709298347933392244255, −4.69789260332584874012026368050, −4.20713022791218069208995118319, −3.98425745809863797388524273556, −3.43733093714784673823352089163, −3.15283535359670198147987758211, −2.56186136651496180776803030509, −1.69543703243622900568843123875, −1.46654260983831209793963632514, −0.68383982668508738324042273792,
0.68383982668508738324042273792, 1.46654260983831209793963632514, 1.69543703243622900568843123875, 2.56186136651496180776803030509, 3.15283535359670198147987758211, 3.43733093714784673823352089163, 3.98425745809863797388524273556, 4.20713022791218069208995118319, 4.69789260332584874012026368050, 5.22767463709298347933392244255, 5.32327541962982309488265821620, 5.87099393414101980963581132704, 6.11287410826067766002001180262, 6.77686020155119560370831250674, 7.16898432497399265152039396491, 7.50949738233287366161816190885, 7.65045826460458957704852268444, 8.062701672845131715446036238713, 8.539400458459226246713354083138, 8.979156002578638622016660934646
