| L(s) = 1 | + 9-s + 8·11-s + 6·19-s + 8·29-s − 14·31-s − 32·41-s − 11·49-s + 20·59-s + 12·61-s + 2·79-s + 4·89-s + 8·99-s − 24·101-s − 6·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 6·171-s + 173-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 2.41·11-s + 1.37·19-s + 1.48·29-s − 2.51·31-s − 4.99·41-s − 1.57·49-s + 2.60·59-s + 1.53·61-s + 0.225·79-s + 0.423·89-s + 0.804·99-s − 2.38·101-s − 0.574·109-s + 3.45·121-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 − 3.53·169-s + 0.458·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.160117348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.160117348\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 6 T^{2} - 2173 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 42 T^{2} - 1045 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 79 T^{2} + 912 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.52980699123046276067391203766, −6.41377950513616464026980102715, −6.19749145680771849829122434297, −5.74543774172179628518830117111, −5.54935496225336096582893441273, −5.36154220020574193797762323702, −5.22091761552953133000725145062, −5.16704232185747018864012424917, −4.65271578112438873857595804407, −4.60781787351233977898374433539, −4.41102754424863926746585231563, −3.86891073758060899551150726913, −3.72099769764717286966548204451, −3.67490527006674942546837791078, −3.58642653378470077180351096855, −3.27637809995456680316790594691, −2.93462618274489109843747650098, −2.61045979519073208635472264401, −2.40460427652870945629897663001, −1.82603767479830336944618780585, −1.63419916026507724319743651807, −1.46374658656284495258965082485, −1.42234121603893939128920430196, −0.63779283047354412179779196576, −0.47882458029656790425452158964,
0.47882458029656790425452158964, 0.63779283047354412179779196576, 1.42234121603893939128920430196, 1.46374658656284495258965082485, 1.63419916026507724319743651807, 1.82603767479830336944618780585, 2.40460427652870945629897663001, 2.61045979519073208635472264401, 2.93462618274489109843747650098, 3.27637809995456680316790594691, 3.58642653378470077180351096855, 3.67490527006674942546837791078, 3.72099769764717286966548204451, 3.86891073758060899551150726913, 4.41102754424863926746585231563, 4.60781787351233977898374433539, 4.65271578112438873857595804407, 5.16704232185747018864012424917, 5.22091761552953133000725145062, 5.36154220020574193797762323702, 5.54935496225336096582893441273, 5.74543774172179628518830117111, 6.19749145680771849829122434297, 6.41377950513616464026980102715, 6.52980699123046276067391203766
