| L(s) = 1 | − 2·4-s + 4·5-s − 4·11-s + 3·16-s + 12·19-s − 8·20-s + 8·25-s + 24·29-s + 16·31-s − 4·41-s + 8·44-s + 8·49-s − 16·55-s − 4·59-s + 8·61-s − 4·64-s − 24·76-s + 12·80-s − 32·89-s + 48·95-s − 16·100-s − 32·101-s + 32·109-s − 48·116-s − 22·121-s − 32·124-s + 20·125-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s − 1.20·11-s + 3/4·16-s + 2.75·19-s − 1.78·20-s + 8/5·25-s + 4.45·29-s + 2.87·31-s − 0.624·41-s + 1.20·44-s + 8/7·49-s − 2.15·55-s − 0.520·59-s + 1.02·61-s − 1/2·64-s − 2.75·76-s + 1.34·80-s − 3.39·89-s + 4.92·95-s − 8/5·100-s − 3.18·101-s + 3.06·109-s − 4.45·116-s − 2·121-s − 2.87·124-s + 1.78·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.716888280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.716888280\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 18 T^{2} + 563 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 6123 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 11378 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8850 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T - 31 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 14043 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1683 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.48608423741367894544219739792, −7.01160054779817175926305125550, −6.87629521328619461622609006081, −6.61128659963265124407326608742, −6.47223270963875518134141871263, −6.16570458790057600351568326608, −5.91199539837986528577194962637, −5.63756084270442273986474525059, −5.48432430691897996587932200732, −5.30256995889491415984418142808, −4.85873904737322064902132315508, −4.80038603688114418582371002768, −4.78810623404596434139534368663, −4.36982706934895730083481196472, −3.92886752719766981947595455177, −3.81770315642419600190342845360, −3.07834342215822356697404443467, −2.96277239923228904556067955980, −2.89097214218450393454757721109, −2.50203818729637776542790619321, −2.48484823064954081762793574373, −1.55285940544217600692876940844, −1.41574081641130370632121772831, −0.867162616904535322279206243995, −0.74853655728891269415823369268,
0.74853655728891269415823369268, 0.867162616904535322279206243995, 1.41574081641130370632121772831, 1.55285940544217600692876940844, 2.48484823064954081762793574373, 2.50203818729637776542790619321, 2.89097214218450393454757721109, 2.96277239923228904556067955980, 3.07834342215822356697404443467, 3.81770315642419600190342845360, 3.92886752719766981947595455177, 4.36982706934895730083481196472, 4.78810623404596434139534368663, 4.80038603688114418582371002768, 4.85873904737322064902132315508, 5.30256995889491415984418142808, 5.48432430691897996587932200732, 5.63756084270442273986474525059, 5.91199539837986528577194962637, 6.16570458790057600351568326608, 6.47223270963875518134141871263, 6.61128659963265124407326608742, 6.87629521328619461622609006081, 7.01160054779817175926305125550, 7.48608423741367894544219739792
