| L(s) = 1 | + 422·9-s + 1.07e3·11-s + 2.22e3·19-s − 5.83e3·29-s + 5.24e3·31-s + 340·41-s − 9.65e3·49-s + 8.29e4·59-s + 3.09e4·61-s − 5.71e4·71-s − 1.38e5·79-s + 1.19e5·81-s + 2.53e5·89-s + 4.52e5·99-s + 1.28e4·101-s − 4.97e3·109-s + 5.39e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.60e5·169-s + ⋯ |
| L(s) = 1 | + 1.73·9-s + 2.67·11-s + 1.41·19-s − 1.28·29-s + 0.980·31-s + 0.0315·41-s − 0.574·49-s + 3.10·59-s + 1.06·61-s − 1.34·71-s − 2.49·79-s + 2.01·81-s + 3.39·89-s + 4.63·99-s + 0.125·101-s − 0.0400·109-s + 3.35·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.702·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.949232791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.949232791\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 422 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 9650 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 536 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 260950 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1206430 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1112 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2530030 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2918 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2624 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49234150 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 170 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 103108298 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 458688990 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 344527302 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 41480 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 15462 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2269936678 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28592 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1265674286 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 69152 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6449241286 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 126806 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13294636414 T^{2} + p^{10} T^{4} \) |
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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
−9.768358916214408601910225410688, −9.340824335920448743518866328526, −8.873201340201468672576508167204, −8.736704950128098198739088242230, −7.79984068717065662182975384321, −7.61120162069170279367033129017, −6.96691888251164231504645833507, −6.81954108633793409855173744391, −6.42990435346206177238637998572, −5.81714848316671794806421781432, −5.30280178646016097095698086359, −4.70968019075414304228255501888, −4.15674443373212159223554124100, −3.89632223517782213448149176344, −3.53144618881840690568479977725, −2.80140835190084112013487313543, −1.77481056147175849265754772774, −1.67900312097751265361853060909, −0.917659211573727876752183211192, −0.73216932103521270983262236313,
0.73216932103521270983262236313, 0.917659211573727876752183211192, 1.67900312097751265361853060909, 1.77481056147175849265754772774, 2.80140835190084112013487313543, 3.53144618881840690568479977725, 3.89632223517782213448149176344, 4.15674443373212159223554124100, 4.70968019075414304228255501888, 5.30280178646016097095698086359, 5.81714848316671794806421781432, 6.42990435346206177238637998572, 6.81954108633793409855173744391, 6.96691888251164231504645833507, 7.61120162069170279367033129017, 7.79984068717065662182975384321, 8.736704950128098198739088242230, 8.873201340201468672576508167204, 9.340824335920448743518866328526, 9.768358916214408601910225410688
