| L(s) = 1 | + 4·3-s + 2·5-s + 4·7-s + 8·9-s − 2·13-s + 8·15-s − 10·17-s + 16·21-s + 4·23-s − 25-s + 12·27-s + 8·29-s + 8·35-s + 2·37-s − 8·39-s + 12·43-s + 16·45-s + 4·47-s + 8·49-s − 40·51-s + 14·53-s + 32·63-s − 4·65-s + 20·67-s + 16·69-s + 6·73-s − 4·75-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s − 0.554·13-s + 2.06·15-s − 2.42·17-s + 3.49·21-s + 0.834·23-s − 1/5·25-s + 2.30·27-s + 1.48·29-s + 1.35·35-s + 0.328·37-s − 1.28·39-s + 1.82·43-s + 2.38·45-s + 0.583·47-s + 8/7·49-s − 5.60·51-s + 1.92·53-s + 4.03·63-s − 0.496·65-s + 2.44·67-s + 1.92·69-s + 0.702·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.649717871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.649717871\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.637445918137488821330463097390, −9.248259783175160853081988788305, −8.984064888830060203854062546556, −8.665726796564575432950957919099, −8.420635345914108875254933326695, −7.971058108206227821063863793780, −7.64084928425883164115914497256, −7.09699356874525161932203036268, −6.72632447822442935177252188295, −6.37260494201619642663893724475, −5.37929589965047189799786155287, −5.32382502031391882329072238250, −4.54233850297811252151142764348, −4.13715078589335826802777805672, −3.99378345328561432670310880824, −2.83453826517783153703154103552, −2.58363013431982497904613150855, −2.37013766845954480417782430793, −1.80007069764934385275119207374, −1.07273876415153275693501838165,
1.07273876415153275693501838165, 1.80007069764934385275119207374, 2.37013766845954480417782430793, 2.58363013431982497904613150855, 2.83453826517783153703154103552, 3.99378345328561432670310880824, 4.13715078589335826802777805672, 4.54233850297811252151142764348, 5.32382502031391882329072238250, 5.37929589965047189799786155287, 6.37260494201619642663893724475, 6.72632447822442935177252188295, 7.09699356874525161932203036268, 7.64084928425883164115914497256, 7.971058108206227821063863793780, 8.420635345914108875254933326695, 8.665726796564575432950957919099, 8.984064888830060203854062546556, 9.248259783175160853081988788305, 9.637445918137488821330463097390
