| L(s) = 1 | − 2.65e3·4-s + 1.15e5·9-s + 3.19e6·16-s + 3.79e6·19-s − 1.19e8·31-s − 3.05e8·36-s + 5.35e8·49-s + 4.12e9·61-s − 1.01e9·64-s − 1.00e10·76-s − 5.95e9·79-s + 9.78e9·81-s + 6.18e10·109-s + 3.39e10·121-s + 3.16e11·124-s + ⋯ |
| L(s) = 1 | − 2.59·4-s + 1.95·9-s + 3.04·16-s + 1.53·19-s − 4.16·31-s − 5.05·36-s + 1.89·49-s + 4.88·61-s − 0.943·64-s − 3.97·76-s − 1.93·79-s + 2.80·81-s + 4.02·109-s + 1.30·121-s + 10.7·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.09515155135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09515155135\) |
| \(L(6)\) |
|
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 | $C_2^2$ | \( 1 - 158 p^{6} T^{2} + p^{20} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 83 p^{4} T^{2} + p^{20} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 5468158 p^{2} T^{2} + p^{20} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 16976849522 T^{2} + p^{20} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 246934503982 T^{2} + p^{20} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 3914170410818 T^{2} + p^{20} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 949462 T + p^{10} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 75790028393378 T^{2} + p^{20} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 831288000078482 T^{2} + p^{20} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 29793118 T + p^{10} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 5919088130907982 T^{2} + p^{20} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 6157996032931678 T^{2} + p^{20} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 31683971389442062 T^{2} + p^{20} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 33376598707262018 T^{2} + p^{20} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 312876100791567218 T^{2} + p^{20} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 600827685707033522 T^{2} + p^{20} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1030793642 T + p^{10} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 123513295387882222 T^{2} + p^{20} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 690030731290713118 T^{2} + p^{20} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 492527192253207262 T^{2} + p^{20} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1488647618 T + p^{10} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 29429698146400299218 T^{2} + p^{20} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 26116812713945754722 T^{2} + p^{20} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - \)\(14\!\cdots\!22\)\( T^{2} + p^{20} T^{4} )^{2} \) |
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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
−8.939269240590191900061756058349, −8.532519692298142264229800425002, −8.044662668082628309486669218072, −7.71694986154623162401623358514, −7.31299900644516150034138962887, −7.15609411428775993318470655971, −7.05871400753479494688909533929, −6.57046203396942314801660869336, −5.79213603636283002756539926444, −5.56987243544534124043646156440, −5.48073696670702644320491819588, −4.99989769121863757152268919451, −4.76018168905146478039834243696, −4.47543584295662966135596811413, −3.89546068302960698887427616834, −3.88563395138249514111228540473, −3.61522447395817311214790603496, −3.34843624702376002260212704011, −2.47853916474887472422562810483, −2.03432631014960821281088394625, −1.84063471934193278240357721186, −1.05761764242447470321250595079, −1.00489256185238326391860917962, −0.67060755178337734554954637385, −0.05189891337487989931363709661,
0.05189891337487989931363709661, 0.67060755178337734554954637385, 1.00489256185238326391860917962, 1.05761764242447470321250595079, 1.84063471934193278240357721186, 2.03432631014960821281088394625, 2.47853916474887472422562810483, 3.34843624702376002260212704011, 3.61522447395817311214790603496, 3.88563395138249514111228540473, 3.89546068302960698887427616834, 4.47543584295662966135596811413, 4.76018168905146478039834243696, 4.99989769121863757152268919451, 5.48073696670702644320491819588, 5.56987243544534124043646156440, 5.79213603636283002756539926444, 6.57046203396942314801660869336, 7.05871400753479494688909533929, 7.15609411428775993318470655971, 7.31299900644516150034138962887, 7.71694986154623162401623358514, 8.044662668082628309486669218072, 8.532519692298142264229800425002, 8.939269240590191900061756058349
