| L(s) = 1 | − 5-s + 2·29-s − 5·37-s − 2·41-s − 4·49-s + 5·53-s + 2·61-s − 3·89-s + 2·101-s + 2·109-s − 5·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 5·185-s + ⋯ |
| L(s) = 1 | − 5-s + 2·29-s − 5·37-s − 2·41-s − 4·49-s + 5·53-s + 2·61-s − 3·89-s + 2·101-s + 2·109-s − 5·113-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 5·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1964750088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1964750088\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.45492659240324156768767701268, −6.00792228712597603146521484902, −5.78974993174937988107251461076, −5.66394339116937220832616683774, −5.19806232696311138521407665250, −5.18231871134800084587725311968, −5.07568582229331805525327098141, −4.97761918195855040222380763838, −4.74679029317903758268557891756, −4.42613765538212853670340185989, −4.08998780944420589025165940570, −3.86325151408085693293601480365, −3.74686920550808368791992454871, −3.69526012511733357314686300071, −3.38560599019448104241631322657, −3.22783399604624739038805683852, −2.89476261929471736936922618621, −2.62670045486089727583432503699, −2.44969324779487460457536327300, −2.05845295932628755071783506641, −1.93400362703778880498376230326, −1.48455746920464051305992032777, −1.22769238624880365053097598561, −1.07918697540379694633948221534, −0.16797970694903501799582829243,
0.16797970694903501799582829243, 1.07918697540379694633948221534, 1.22769238624880365053097598561, 1.48455746920464051305992032777, 1.93400362703778880498376230326, 2.05845295932628755071783506641, 2.44969324779487460457536327300, 2.62670045486089727583432503699, 2.89476261929471736936922618621, 3.22783399604624739038805683852, 3.38560599019448104241631322657, 3.69526012511733357314686300071, 3.74686920550808368791992454871, 3.86325151408085693293601480365, 4.08998780944420589025165940570, 4.42613765538212853670340185989, 4.74679029317903758268557891756, 4.97761918195855040222380763838, 5.07568582229331805525327098141, 5.18231871134800084587725311968, 5.19806232696311138521407665250, 5.66394339116937220832616683774, 5.78974993174937988107251461076, 6.00792228712597603146521484902, 6.45492659240324156768767701268
