| L(s) = 1 | + 4·7-s + 2·13-s − 4·19-s + 12·23-s + 4·25-s + 28·31-s − 4·37-s − 2·43-s − 12·47-s − 6·49-s + 14·61-s − 14·67-s − 12·71-s + 14·73-s + 10·79-s − 48·83-s − 12·89-s + 8·91-s + 8·97-s + 12·101-s − 44·103-s − 24·107-s − 16·109-s + 24·113-s − 32·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.554·13-s − 0.917·19-s + 2.50·23-s + 4/5·25-s + 5.02·31-s − 0.657·37-s − 0.304·43-s − 1.75·47-s − 6/7·49-s + 1.79·61-s − 1.71·67-s − 1.42·71-s + 1.63·73-s + 1.12·79-s − 5.26·83-s − 1.27·89-s + 0.838·91-s + 0.812·97-s + 1.19·101-s − 4.33·103-s − 2.32·107-s − 1.53·109-s + 2.25·113-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086178641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086178641\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 68 T^{2} - 360 T^{3} + 1935 T^{4} - 360 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 34 T^{2} + 315 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 14 T + 105 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T - 21 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 77 T^{2} - 10 T^{3} + 4540 T^{4} - 10 p T^{5} - 77 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 100 T^{2} + 7191 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 64 T^{2} + 615 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 49 T^{2} - 350 T^{3} + 5932 T^{4} - 350 p T^{5} + 49 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 19 T^{2} + 602 T^{3} + 13708 T^{4} + 602 p T^{5} + 19 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10 T - 29 T^{2} + 290 T^{3} + 2500 T^{4} + 290 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 24 T + 286 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 64 T^{2} + 360 T^{3} + 22527 T^{4} + 360 p T^{5} - 64 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 70 T^{2} + 1600 T^{3} - 15581 T^{4} + 1600 p T^{5} + 70 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.32622890429612200343865429299, −6.32070748877121970874154005619, −5.84230816844695744911190660133, −5.37297991203305265203204394910, −5.35469010939283095817886121049, −5.18162452601709272266897120162, −5.05447872020811688752226751843, −4.98032411430193180689358283992, −4.50862223496521088428999025979, −4.28523323120829915625205039347, −4.28509820363736693292286004017, −4.21436233724926755625181091434, −3.72814264786763131287996927548, −3.53613670243333316635878370147, −2.93445201699375731243569214768, −2.90662205789411750135157032890, −2.82366139953971022098758605595, −2.72738561774683117188265542442, −2.37891410509404473329934743639, −1.80166654379551554099750414297, −1.54040320242911875023823481627, −1.35220244792899814900131375031, −1.10817947382482915922531723034, −1.00590594728290876301023933007, −0.19829702730117330490618636113,
0.19829702730117330490618636113, 1.00590594728290876301023933007, 1.10817947382482915922531723034, 1.35220244792899814900131375031, 1.54040320242911875023823481627, 1.80166654379551554099750414297, 2.37891410509404473329934743639, 2.72738561774683117188265542442, 2.82366139953971022098758605595, 2.90662205789411750135157032890, 2.93445201699375731243569214768, 3.53613670243333316635878370147, 3.72814264786763131287996927548, 4.21436233724926755625181091434, 4.28509820363736693292286004017, 4.28523323120829915625205039347, 4.50862223496521088428999025979, 4.98032411430193180689358283992, 5.05447872020811688752226751843, 5.18162452601709272266897120162, 5.35469010939283095817886121049, 5.37297991203305265203204394910, 5.84230816844695744911190660133, 6.32070748877121970874154005619, 6.32622890429612200343865429299
