| L(s) = 1 | + 2·5-s − 4·7-s + 2·11-s − 8·19-s + 3·25-s − 4·29-s − 4·31-s − 8·35-s + 4·37-s − 4·41-s − 4·43-s − 8·47-s − 4·53-s + 4·55-s − 12·59-s + 12·61-s − 4·71-s − 8·77-s − 16·79-s − 12·83-s − 4·89-s − 16·95-s + 12·97-s − 20·101-s − 12·107-s − 4·109-s − 4·113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.603·11-s − 1.83·19-s + 3/5·25-s − 0.742·29-s − 0.718·31-s − 1.35·35-s + 0.657·37-s − 0.624·41-s − 0.609·43-s − 1.16·47-s − 0.549·53-s + 0.539·55-s − 1.56·59-s + 1.53·61-s − 0.474·71-s − 0.911·77-s − 1.80·79-s − 1.31·83-s − 0.423·89-s − 1.64·95-s + 1.21·97-s − 1.99·101-s − 1.16·107-s − 0.383·109-s − 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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
−8.194355030886597087753601489674, −8.163684674078997994178342669140, −7.24642568476115040161137961396, −7.24061560018978877647641821837, −6.61677892911359614889753164693, −6.40803401000670299423274467730, −6.16696095168987411800175414965, −5.90088373427917014200973283116, −5.18830216875547986691653458756, −5.10186981458566473880135099115, −4.26631656801897963867301040378, −4.17669831550017246343552535878, −3.51168180516876892914451575964, −3.26746097291734380070123010035, −2.64410486088545206910864162385, −2.35306552808578843653954551269, −1.56632787403980689866663588515, −1.41389604083740205549666632957, 0, 0,
1.41389604083740205549666632957, 1.56632787403980689866663588515, 2.35306552808578843653954551269, 2.64410486088545206910864162385, 3.26746097291734380070123010035, 3.51168180516876892914451575964, 4.17669831550017246343552535878, 4.26631656801897963867301040378, 5.10186981458566473880135099115, 5.18830216875547986691653458756, 5.90088373427917014200973283116, 6.16696095168987411800175414965, 6.40803401000670299423274467730, 6.61677892911359614889753164693, 7.24061560018978877647641821837, 7.24642568476115040161137961396, 8.163684674078997994178342669140, 8.194355030886597087753601489674
