| L(s) = 1 | + 4·4-s + 2·9-s − 12·11-s + 4·16-s − 12·23-s + 10·25-s + 8·36-s − 4·37-s − 48·44-s − 4·49-s − 16·64-s + 44·67-s + 36·71-s − 15·81-s − 48·92-s − 24·99-s + 40·100-s − 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s − 16·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2/3·9-s − 3.61·11-s + 16-s − 2.50·23-s + 2·25-s + 4/3·36-s − 0.657·37-s − 7.23·44-s − 4/7·49-s − 2·64-s + 5.37·67-s + 4.27·71-s − 5/3·81-s − 5.00·92-s − 2.41·99-s + 4·100-s − 1.12·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9834270573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9834270573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 173 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−11.02392676508492289876608055419, −10.33685579134013357063090394577, −10.28226613420318595849526213367, −10.12998230872479206542644840832, −9.864859786780334358578785974913, −9.395875319706378008010643248436, −8.951487388338861586861358538532, −8.414570966927381454188131422040, −8.072187683008054892937878537238, −8.053007278593077786933402589079, −7.76468001337108502788456218067, −7.34063380060344316724817172191, −6.92519303809312994562885024199, −6.82062635626822745794787763240, −6.34346893034830953426613045021, −6.20543042379025854503750523003, −5.29394234540745949716918476449, −5.28719110765115007953962567184, −5.15122160219172806297229899427, −4.41761480213862959022213035276, −3.82069300975700524479502475637, −3.25780717647641430075884979789, −2.48357509766793217070371542497, −2.47464235229053040124719550329, −2.01979387140989027606721966788,
2.01979387140989027606721966788, 2.47464235229053040124719550329, 2.48357509766793217070371542497, 3.25780717647641430075884979789, 3.82069300975700524479502475637, 4.41761480213862959022213035276, 5.15122160219172806297229899427, 5.28719110765115007953962567184, 5.29394234540745949716918476449, 6.20543042379025854503750523003, 6.34346893034830953426613045021, 6.82062635626822745794787763240, 6.92519303809312994562885024199, 7.34063380060344316724817172191, 7.76468001337108502788456218067, 8.053007278593077786933402589079, 8.072187683008054892937878537238, 8.414570966927381454188131422040, 8.951487388338861586861358538532, 9.395875319706378008010643248436, 9.864859786780334358578785974913, 10.12998230872479206542644840832, 10.28226613420318595849526213367, 10.33685579134013357063090394577, 11.02392676508492289876608055419
