| L(s) = 1 | + 2·3-s + 2·9-s − 8·16-s − 18·23-s − 4·27-s + 14·37-s + 24·47-s − 16·48-s + 12·53-s − 26·67-s − 36·69-s − 12·71-s − 3·81-s + 34·97-s − 8·103-s + 28·111-s + 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s − 16·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2/3·9-s − 2·16-s − 3.75·23-s − 0.769·27-s + 2.30·37-s + 3.50·47-s − 2.30·48-s + 1.64·53-s − 3.17·67-s − 4.33·69-s − 1.42·71-s − 1/3·81-s + 3.45·97-s − 0.788·103-s + 2.65·111-s + 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s − 4/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.934231004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934231004\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} ) \) |
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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
−8.605911567743401623460785649784, −8.598444109599867019624350184233, −8.224861804558758118498998955970, −7.76504148124381470473009540777, −7.64822811626934895706330978836, −7.38512592330015777052992705450, −7.28870051174134250243606648158, −7.13927080096932681629367234109, −6.29549427304380644602304140222, −6.20610660741576856452017853664, −6.19595139753805814102024382613, −5.72761969607071608942519538542, −5.69043508825970887996231562902, −5.03016936984670213051871341209, −4.55715787872519691070562717998, −4.44350123808842935514552351248, −4.06142779542640783092370028368, −4.03507083676076211079292615487, −3.61842416069366211893682239325, −3.08289582069765628488987595498, −2.64116739921502670105264484467, −2.33299897427951498499700203720, −2.05852293863090282671698647731, −1.77828205882899532052772551327, −0.60601157062078727803854975180,
0.60601157062078727803854975180, 1.77828205882899532052772551327, 2.05852293863090282671698647731, 2.33299897427951498499700203720, 2.64116739921502670105264484467, 3.08289582069765628488987595498, 3.61842416069366211893682239325, 4.03507083676076211079292615487, 4.06142779542640783092370028368, 4.44350123808842935514552351248, 4.55715787872519691070562717998, 5.03016936984670213051871341209, 5.69043508825970887996231562902, 5.72761969607071608942519538542, 6.19595139753805814102024382613, 6.20610660741576856452017853664, 6.29549427304380644602304140222, 7.13927080096932681629367234109, 7.28870051174134250243606648158, 7.38512592330015777052992705450, 7.64822811626934895706330978836, 7.76504148124381470473009540777, 8.224861804558758118498998955970, 8.598444109599867019624350184233, 8.605911567743401623460785649784
