| L(s) = 1 | − 8·4-s + 6·9-s + 40·16-s − 48·36-s − 11·49-s − 160·64-s + 16·79-s + 27·81-s − 76·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 240·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 88·196-s + ⋯ |
| L(s) = 1 | − 4·4-s + 2·9-s + 10·16-s − 8·36-s − 1.57·49-s − 20·64-s + 1.80·79-s + 3·81-s − 7.27·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 44/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5139687587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5139687587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.892373657994132158383364893483, −7.72472803048816042952177769385, −7.52691468897359460854277318247, −7.31909911075668038044554405328, −6.94528768279055183720331696645, −6.48180853456505477535957617944, −6.27331866249875387612168106742, −6.26650531315286869363913951717, −5.65537697477364257038411022252, −5.34934078652998875859280284817, −5.25319537156480419360196590242, −5.05521707132237474578327744846, −4.69970713442226506322571479400, −4.65708942210780292188596659207, −4.23950650292507199390242448515, −3.97788750817692412371426635244, −3.95588915906764318521666195230, −3.63816324703329082798230012468, −3.30527053522407803150577570583, −2.98904242763748093802211547049, −2.40628662692044119453696030807, −1.65346617475158190382470573438, −1.32405775916972774205726664311, −1.02685279558057878079671248462, −0.33090966372630186139671030315,
0.33090966372630186139671030315, 1.02685279558057878079671248462, 1.32405775916972774205726664311, 1.65346617475158190382470573438, 2.40628662692044119453696030807, 2.98904242763748093802211547049, 3.30527053522407803150577570583, 3.63816324703329082798230012468, 3.95588915906764318521666195230, 3.97788750817692412371426635244, 4.23950650292507199390242448515, 4.65708942210780292188596659207, 4.69970713442226506322571479400, 5.05521707132237474578327744846, 5.25319537156480419360196590242, 5.34934078652998875859280284817, 5.65537697477364257038411022252, 6.26650531315286869363913951717, 6.27331866249875387612168106742, 6.48180853456505477535957617944, 6.94528768279055183720331696645, 7.31909911075668038044554405328, 7.52691468897359460854277318247, 7.72472803048816042952177769385, 7.892373657994132158383364893483
