| L(s) = 1 | + 6·17-s + 2·19-s + 2·23-s + 12·31-s − 12·47-s − 5·49-s − 8·53-s + 26·61-s + 10·79-s − 32·83-s − 26·107-s + 6·109-s + 56·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.45·17-s + 0.458·19-s + 0.417·23-s + 2.15·31-s − 1.75·47-s − 5/7·49-s − 1.09·53-s + 3.32·61-s + 1.12·79-s − 3.51·83-s − 2.51·107-s + 0.574·109-s + 5.26·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.190594769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.190594769\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - T + 32 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 2750 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 5214 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 13 T + 150 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 224 T^{2} + 21294 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 240 T^{2} + 24254 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 245 T^{2} + 25308 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 5 T + 150 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 19680 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.96395781570347091367448424927, −5.41766406762756637080804199645, −5.32789305805261204759040745575, −5.23300747699282763918453313739, −5.07045460396505011138517434898, −4.83054020413357708589665482605, −4.48483382057202042882696680291, −4.45810804702719802146526326840, −4.35278035397212281253843900831, −3.94639786941955030932510199368, −3.76046235336851797048802232346, −3.52921915070968709895232904842, −3.45789911094042828983009957253, −2.94329473706845381631395099830, −2.89325901103071274344096025205, −2.86927544612343173025340892033, −2.79596926315248789199287762356, −2.09326986342926878530499265156, −1.95177755798862582669325360016, −1.78185503762988069872228197376, −1.58032710934954147072065298929, −1.13360164366480695479661551022, −0.861216989420709836613074520323, −0.65048520006419534188770371262, −0.40064481440566365935032534760,
0.40064481440566365935032534760, 0.65048520006419534188770371262, 0.861216989420709836613074520323, 1.13360164366480695479661551022, 1.58032710934954147072065298929, 1.78185503762988069872228197376, 1.95177755798862582669325360016, 2.09326986342926878530499265156, 2.79596926315248789199287762356, 2.86927544612343173025340892033, 2.89325901103071274344096025205, 2.94329473706845381631395099830, 3.45789911094042828983009957253, 3.52921915070968709895232904842, 3.76046235336851797048802232346, 3.94639786941955030932510199368, 4.35278035397212281253843900831, 4.45810804702719802146526326840, 4.48483382057202042882696680291, 4.83054020413357708589665482605, 5.07045460396505011138517434898, 5.23300747699282763918453313739, 5.32789305805261204759040745575, 5.41766406762756637080804199645, 5.96395781570347091367448424927
