| L(s) = 1 | − 2·7-s − 3·9-s − 2·19-s − 22·31-s + 22·37-s + 42·43-s − 13·49-s + 6·63-s − 32·67-s + 14·73-s + 10·97-s − 34·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 6·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s − 0.458·19-s − 3.95·31-s + 3.61·37-s + 6.40·43-s − 1.85·49-s + 0.755·63-s − 3.90·67-s + 1.63·73-s + 1.01·97-s − 3.25·109-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-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 − 1.76·169-s + 0.458·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.355034381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355034381\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 11 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 97 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−7.69102523517801316541474215193, −7.53955333426378639849482794847, −7.14894865048505321178731596140, −7.01463811978757149918190596607, −6.76771312757655289382450785896, −6.28446151216042068908027111106, −6.10876017643576487075067734114, −5.96314822524090783097127425888, −5.73089768310786501243845979791, −5.62707708604292567939862738929, −5.45461775003146918560937466510, −4.95284056954874525983196800956, −4.48273828754009542052925205825, −4.35589099404687414300939166953, −4.27416567008580165361832457270, −3.75781921212278492290750664748, −3.74001532546052356122000976691, −3.07908729957842005481963600489, −2.87968410847943133397022746025, −2.84601169693423664069933231679, −2.18071416852696342512453878137, −2.15898954704689304545386044240, −1.54553894668045087705020238489, −0.904864748089091239836475274536, −0.40656572516366267230471561882,
0.40656572516366267230471561882, 0.904864748089091239836475274536, 1.54553894668045087705020238489, 2.15898954704689304545386044240, 2.18071416852696342512453878137, 2.84601169693423664069933231679, 2.87968410847943133397022746025, 3.07908729957842005481963600489, 3.74001532546052356122000976691, 3.75781921212278492290750664748, 4.27416567008580165361832457270, 4.35589099404687414300939166953, 4.48273828754009542052925205825, 4.95284056954874525983196800956, 5.45461775003146918560937466510, 5.62707708604292567939862738929, 5.73089768310786501243845979791, 5.96314822524090783097127425888, 6.10876017643576487075067734114, 6.28446151216042068908027111106, 6.76771312757655289382450785896, 7.01463811978757149918190596607, 7.14894865048505321178731596140, 7.53955333426378639849482794847, 7.69102523517801316541474215193
