| L(s) = 1 | − 6·3-s + 27·9-s − 180·17-s + 232·19-s − 142·25-s − 108·27-s + 108·41-s + 40·43-s − 674·49-s + 1.08e3·51-s − 1.39e3·57-s + 648·59-s − 232·67-s − 2.21e3·73-s + 852·75-s + 405·81-s − 2.30e3·83-s − 1.83e3·89-s + 380·97-s − 504·107-s − 4.42e3·113-s − 2.66e3·121-s − 648·123-s + 127-s − 240·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.56·17-s + 2.80·19-s − 1.13·25-s − 0.769·27-s + 0.411·41-s + 0.141·43-s − 1.96·49-s + 2.96·51-s − 3.23·57-s + 1.42·59-s − 0.423·67-s − 3.54·73-s + 1.31·75-s + 5/9·81-s − 3.04·83-s − 2.18·89-s + 0.397·97-s − 0.455·107-s − 3.68·113-s − 2·121-s − 0.475·123-s + 0.000698·127-s − 0.163·129-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 674 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1322 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 13534 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18722 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 31246 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 100106 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 51694 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 59182 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 123290 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 497666 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1106 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 963890 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1152 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 918 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.608973637026612454610298084311, −9.496020693762901509601645384804, −8.972122515411266132223183645087, −8.460696333250665779317620920861, −7.88793090850533178523831913072, −7.43935419923692050033063350158, −7.00269658465264802776615138857, −6.71342733581175856866836766737, −6.13595186029342502172068421409, −5.66971950052503201187626135694, −5.29569384777169722436055491474, −4.84854109345611160543512489700, −4.10924732652665987726774534048, −4.05900479143116480280775166821, −2.98253769503651987511559744771, −2.60590554348838389545866181964, −1.56513226167443779892859890300, −1.25966626201570182391997511822, 0, 0,
1.25966626201570182391997511822, 1.56513226167443779892859890300, 2.60590554348838389545866181964, 2.98253769503651987511559744771, 4.05900479143116480280775166821, 4.10924732652665987726774534048, 4.84854109345611160543512489700, 5.29569384777169722436055491474, 5.66971950052503201187626135694, 6.13595186029342502172068421409, 6.71342733581175856866836766737, 7.00269658465264802776615138857, 7.43935419923692050033063350158, 7.88793090850533178523831913072, 8.460696333250665779317620920861, 8.972122515411266132223183645087, 9.496020693762901509601645384804, 9.608973637026612454610298084311
