L(s) = 1 | + 2·2-s − 3-s + 4-s − 2·6-s + 4·7-s − 2·8-s + 9-s − 6·11-s − 12-s − 5·13-s + 8·14-s − 4·16-s + 3·17-s + 2·18-s − 14·19-s − 4·21-s − 12·22-s + 2·24-s − 10·26-s + 4·27-s + 4·28-s − 9·29-s − 10·31-s − 2·32-s + 6·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.816·6-s + 1.51·7-s − 0.707·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 1.38·13-s + 2.13·14-s − 16-s + 0.727·17-s + 0.471·18-s − 3.21·19-s − 0.872·21-s − 2.55·22-s + 0.408·24-s − 1.96·26-s + 0.769·27-s + 0.755·28-s − 1.67·29-s − 1.79·31-s − 0.353·32-s + 1.04·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2402900443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2402900443\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
good | 3 | $D_4\times C_2$ | \( 1 + T - 5 T^{3} - 11 T^{4} - 5 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T - 2 T^{2} + 5 T^{3} + 235 T^{4} + 5 p T^{5} - 2 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T - 22 T^{2} + 9 T^{3} + 519 T^{4} + 9 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 25 T^{2} + 96 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T + 8 T^{2} + 135 T^{3} + 2139 T^{4} + 135 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9 T + 26 T^{2} + 243 T^{3} - 1977 T^{4} + 243 p T^{5} + 26 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 7 T - 44 T^{2} - 49 T^{3} + 4693 T^{4} - 49 p T^{5} - 44 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 450 T^{3} - 3540 T^{4} - 450 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 85 T^{2} + 4416 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 97 T^{2} + 5928 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 11 T + 16 T^{2} + 187 T^{3} - 77 T^{4} + 187 p T^{5} + 16 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22 T + 250 T^{2} - 2200 T^{3} + 17839 T^{4} - 2200 p T^{5} + 250 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 121 T^{2} + 9600 T^{4} - 121 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - T - 140 T^{2} + 5 T^{3} + 14479 T^{4} + 5 p T^{5} - 140 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T + 10 T^{2} - 833 T^{3} - 8591 T^{4} - 833 p T^{5} + 10 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 3 T + 163 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 157 T^{2} + 16728 T^{4} - 157 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 11 T - 56 T^{2} - 187 T^{3} + 10183 T^{4} - 187 p T^{5} - 56 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.01675482629958724454542097448, −6.94684056497453821692421842516, −6.87732663761693864358848991356, −6.28397390067160081268627346393, −6.19375803704122686745697877803, −6.06714945522013352095023780173, −5.61871130393530834677856707584, −5.37474007567763474102953803209, −5.30827631537671279337975948631, −5.18729554611703105766497300928, −4.97108515122886910566253515900, −4.57652121498817056118746524337, −4.34493216790591419390008030818, −4.31151990615729356646428985390, −4.13900457145854542046138268306, −3.57416520509140839848254950579, −3.51908579234288403140996210177, −2.89791544025812513703614700364, −2.89275634267949812011236203249, −2.23854681519596534392561530859, −2.23301595935747769826021735541, −1.96308033114466328943785310181, −1.61243259455048431009984848406, −0.848574720259206972900786448466, −0.10217185655563589183540727933,
0.10217185655563589183540727933, 0.848574720259206972900786448466, 1.61243259455048431009984848406, 1.96308033114466328943785310181, 2.23301595935747769826021735541, 2.23854681519596534392561530859, 2.89275634267949812011236203249, 2.89791544025812513703614700364, 3.51908579234288403140996210177, 3.57416520509140839848254950579, 4.13900457145854542046138268306, 4.31151990615729356646428985390, 4.34493216790591419390008030818, 4.57652121498817056118746524337, 4.97108515122886910566253515900, 5.18729554611703105766497300928, 5.30827631537671279337975948631, 5.37474007567763474102953803209, 5.61871130393530834677856707584, 6.06714945522013352095023780173, 6.19375803704122686745697877803, 6.28397390067160081268627346393, 6.87732663761693864358848991356, 6.94684056497453821692421842516, 7.01675482629958724454542097448