| L(s) = 1 | − 2·4-s + 9-s + 3·16-s + 2·29-s − 2·36-s − 2·41-s + 49-s − 2·61-s − 4·64-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 3·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s − 4·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s + 9-s + 3·16-s + 2·29-s − 2·36-s − 2·41-s + 49-s − 2·61-s − 4·64-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 3·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s − 4·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3898857644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3898857644\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.022520811785433525116451494462, −8.006120872951264265066786573584, −7.903753242028869966867672037200, −7.37778346034146958242057896319, −7.17943912351348497531709641840, −6.90353233914736342663985772655, −6.88949502527873404401751038917, −6.26445790760845398235712485742, −6.17075501779488516761979415131, −5.84100056221284411154529286528, −5.75840755322082570063539996459, −5.18998964462435297827539165020, −4.99265802712363048191117049820, −4.91386951587758560842826163324, −4.48356678306683325587443462193, −4.37510281737693269411401181871, −4.24592654490052201150390067501, −3.66735397126259653442367922964, −3.49234463308560482312565558060, −3.10704401620495720348084907954, −3.07305886590573934492989252575, −2.25667130418092342522405004614, −2.00692432004278416078375153215, −1.25276422819773355308339692320, −1.06921259929186878510600224317,
1.06921259929186878510600224317, 1.25276422819773355308339692320, 2.00692432004278416078375153215, 2.25667130418092342522405004614, 3.07305886590573934492989252575, 3.10704401620495720348084907954, 3.49234463308560482312565558060, 3.66735397126259653442367922964, 4.24592654490052201150390067501, 4.37510281737693269411401181871, 4.48356678306683325587443462193, 4.91386951587758560842826163324, 4.99265802712363048191117049820, 5.18998964462435297827539165020, 5.75840755322082570063539996459, 5.84100056221284411154529286528, 6.17075501779488516761979415131, 6.26445790760845398235712485742, 6.88949502527873404401751038917, 6.90353233914736342663985772655, 7.17943912351348497531709641840, 7.37778346034146958242057896319, 7.903753242028869966867672037200, 8.006120872951264265066786573584, 8.022520811785433525116451494462
