| L(s) = 1 | − 3·9-s − 4·11-s − 6·29-s + 4·31-s + 10·41-s + 16·59-s + 14·61-s + 16·71-s − 8·79-s − 26·89-s + 12·99-s − 6·101-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s − 1.11·29-s + 0.718·31-s + 1.56·41-s + 2.08·59-s + 1.79·61-s + 1.89·71-s − 0.900·79-s − 2.75·89-s + 1.20·99-s − 0.597·101-s − 1.72·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122010517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122010517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.927606384832135505937107537569, −8.244832434877401667982860938815, −7.75819461750138587573321806343, −7.38036159906017304798779246382, −7.11633872769919272074151889212, −6.48257535512681288714544784869, −6.42463364161927056867041349096, −5.64733386520496298938240457734, −5.61406359241653358984327465588, −5.20094445439128630483935361484, −4.97783158302393580326857325548, −4.24254958085369080754719515782, −3.86902306078908157210361053440, −3.70111564415404097309701575085, −2.83743416889672640116737305572, −2.57398002177700183351243545641, −2.48718570798707084033580842034, −1.68351657563366189079604723233, −1.00397152571148135835919303822, −0.32058553311134252531728694303,
0.32058553311134252531728694303, 1.00397152571148135835919303822, 1.68351657563366189079604723233, 2.48718570798707084033580842034, 2.57398002177700183351243545641, 2.83743416889672640116737305572, 3.70111564415404097309701575085, 3.86902306078908157210361053440, 4.24254958085369080754719515782, 4.97783158302393580326857325548, 5.20094445439128630483935361484, 5.61406359241653358984327465588, 5.64733386520496298938240457734, 6.42463364161927056867041349096, 6.48257535512681288714544784869, 7.11633872769919272074151889212, 7.38036159906017304798779246382, 7.75819461750138587573321806343, 8.244832434877401667982860938815, 8.927606384832135505937107537569
