| L(s) = 1 | − 3·2-s + 2·3-s + 3·4-s − 6·6-s + 6·7-s + 7·9-s + 6·12-s + 4·13-s − 18·14-s − 2·16-s − 21·18-s + 6·19-s + 12·21-s − 12·26-s + 22·27-s + 18·28-s − 6·32-s + 21·36-s − 18·38-s + 8·39-s − 36·42-s + 12·43-s − 4·48-s + 7·49-s + 12·52-s − 12·53-s − 66·54-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 3/2·4-s − 2.44·6-s + 2.26·7-s + 7/3·9-s + 1.73·12-s + 1.10·13-s − 4.81·14-s − 1/2·16-s − 4.94·18-s + 1.37·19-s + 2.61·21-s − 2.35·26-s + 4.23·27-s + 3.40·28-s − 1.06·32-s + 7/2·36-s − 2.91·38-s + 1.28·39-s − 5.55·42-s + 1.82·43-s − 0.577·48-s + 49-s + 1.66·52-s − 1.64·53-s − 8.98·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.686598213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.686598213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 15 T^{2} + 104 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 13 T^{2} - 120 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 25 T^{2} + 96 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 37 T^{2} + 528 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 1182 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 11 T^{2} - 1248 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 75 T^{2} + 3944 T^{4} + 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5990 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 30 T + 465 T^{2} - 4950 T^{3} + 41444 T^{4} - 4950 p T^{5} + 465 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 7 T^{2} - 180 T^{3} + 6264 T^{4} - 180 p T^{5} + 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 24 T + 311 T^{2} - 2856 T^{3} + 22536 T^{4} - 2856 p T^{5} + 311 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 702 T^{3} + 9500 T^{4} - 702 p T^{5} + 129 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 18734 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 411 T^{2} + 5256 T^{3} + 57128 T^{4} + 5256 p T^{5} + 411 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 191 T^{2} - 1716 T^{3} + 15696 T^{4} - 1716 p T^{5} + 191 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.439062313850353571922820828390, −8.366493690905622219781127877212, −8.157835554837272397313791968929, −7.930541021270174273005203443395, −7.61688080187266661322958541001, −7.17407283803751935450407346878, −7.16083055666735661037394943812, −7.05640886638769359154761955870, −6.55684638700484600407611461967, −6.41542147908600076254446295403, −5.67344096899098389363970767558, −5.52675827295297150310198933980, −5.26231182939316921901199899126, −4.92948970882041061306202912812, −4.61881381391856813205403898449, −4.41246719101620541569166528726, −3.85594446913476065813495661405, −3.78109682012003773829456612159, −3.50262229570791818763406371155, −2.82305398652931461650624333145, −2.25491965541077868038895123401, −2.25476782647259784392907809547, −1.34432894977297429380586293851, −1.20324617796095240324644863891, −1.08853663304724741854885715327,
1.08853663304724741854885715327, 1.20324617796095240324644863891, 1.34432894977297429380586293851, 2.25476782647259784392907809547, 2.25491965541077868038895123401, 2.82305398652931461650624333145, 3.50262229570791818763406371155, 3.78109682012003773829456612159, 3.85594446913476065813495661405, 4.41246719101620541569166528726, 4.61881381391856813205403898449, 4.92948970882041061306202912812, 5.26231182939316921901199899126, 5.52675827295297150310198933980, 5.67344096899098389363970767558, 6.41542147908600076254446295403, 6.55684638700484600407611461967, 7.05640886638769359154761955870, 7.16083055666735661037394943812, 7.17407283803751935450407346878, 7.61688080187266661322958541001, 7.930541021270174273005203443395, 8.157835554837272397313791968929, 8.366493690905622219781127877212, 8.439062313850353571922820828390
