| L(s) = 1 | − 2·9-s − 12·11-s − 14·19-s + 2·29-s − 20·31-s − 4·41-s − 4·49-s − 2·59-s + 14·61-s − 30·71-s − 2·79-s + 9·81-s + 34·89-s + 24·99-s − 22·101-s + 14·109-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 3.61·11-s − 3.21·19-s + 0.371·29-s − 3.59·31-s − 0.624·41-s − 4/7·49-s − 0.260·59-s + 1.79·61-s − 3.56·71-s − 0.225·79-s + 81-s + 3.60·89-s + 2.41·99-s − 2.18·101-s + 1.34·109-s + 4.18·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2052700009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2052700009\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 2 T^{2} - 5325 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.53778704800134409740544086563, −6.33037633272993282165111450303, −6.10714371964561215871727989540, −5.77153856994110900655094108920, −5.56379409686943330221828030992, −5.52780305382413688666017398860, −5.51834184769869826589510585644, −4.92898894419454928772284863214, −4.91096878541930056402387134607, −4.77308700900269260062626641040, −4.37702454940207645751945942305, −4.12841700320055809496326994545, −4.11532198377796116124187267334, −3.42111802997040223553139115874, −3.39158629347785118188882476289, −3.32294012882191153578081542509, −2.92295046690702591822291711862, −2.47654431552578436207129508419, −2.38117468400252566111854659720, −2.12914862500964996315940573592, −2.07558944369751907460501090615, −1.67857715275242319453313053644, −1.15936733279726361671232939108, −0.31738247707135691938915555307, −0.18587966166460704873263648031,
0.18587966166460704873263648031, 0.31738247707135691938915555307, 1.15936733279726361671232939108, 1.67857715275242319453313053644, 2.07558944369751907460501090615, 2.12914862500964996315940573592, 2.38117468400252566111854659720, 2.47654431552578436207129508419, 2.92295046690702591822291711862, 3.32294012882191153578081542509, 3.39158629347785118188882476289, 3.42111802997040223553139115874, 4.11532198377796116124187267334, 4.12841700320055809496326994545, 4.37702454940207645751945942305, 4.77308700900269260062626641040, 4.91096878541930056402387134607, 4.92898894419454928772284863214, 5.51834184769869826589510585644, 5.52780305382413688666017398860, 5.56379409686943330221828030992, 5.77153856994110900655094108920, 6.10714371964561215871727989540, 6.33037633272993282165111450303, 6.53778704800134409740544086563
