| L(s) = 1 | + 240·4-s + 2.59e4·9-s − 5.10e4·11-s − 2.04e5·16-s + 7.00e5·19-s − 1.34e7·29-s − 1.28e7·31-s + 6.21e6·36-s − 2.04e7·41-s − 1.22e7·44-s − 5.76e6·49-s − 1.69e8·59-s + 2.93e7·61-s − 1.12e8·64-s + 1.23e8·71-s + 1.68e8·76-s − 5.52e8·79-s + 2.83e8·81-s + 1.79e9·89-s − 1.32e9·99-s − 2.92e9·101-s + 2.85e9·109-s − 3.22e9·116-s − 2.75e9·121-s − 3.07e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 0.468·4-s + 1.31·9-s − 1.05·11-s − 0.780·16-s + 1.23·19-s − 3.52·29-s − 2.49·31-s + 0.617·36-s − 1.13·41-s − 0.493·44-s − 1/7·49-s − 1.82·59-s + 0.271·61-s − 0.834·64-s + 0.578·71-s + 0.577·76-s − 1.59·79-s + 0.732·81-s + 3.02·89-s − 1.38·99-s − 2.79·101-s + 1.93·109-s − 1.65·116-s − 1.16·121-s − 1.16·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.004620846595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004620846595\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 p^{4} T^{2} + p^{18} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 25910 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 25548 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 19419201110 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 39860147970 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 350060 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3215451178350 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6720430 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6412208 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 254551829737030 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10224678 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 98331986961670 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1679179082717790 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3317078981240550 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 84934780 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14677822 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5395454617637450 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 61901952 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 37221321223132750 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 276107480 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 368552100942726870 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 896368470 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 66485378912592350 T^{2} + p^{18} T^{4} \) |
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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
−11.77818916129660758820564015951, −10.91698949898015833804265695104, −10.41303416479171704547848240963, −9.604424521197358853941123539827, −9.416358635380412097278021659071, −8.985528234511624577613196419023, −7.971298553194077500011485821295, −7.61037842284467470986765016175, −7.12309892097355287467949277125, −6.98041080334226520259682781715, −5.91664450131519930752339822026, −5.49651839571519605956052754671, −5.00703186109405715450804645380, −4.27160892903382033168473247630, −3.57546077147594006242642379311, −3.20432161263086154664020240966, −2.04611581031401876354428720246, −1.93316565915546368717561474640, −1.23242893207099596914871646219, −0.01446230521967327860851920175,
0.01446230521967327860851920175, 1.23242893207099596914871646219, 1.93316565915546368717561474640, 2.04611581031401876354428720246, 3.20432161263086154664020240966, 3.57546077147594006242642379311, 4.27160892903382033168473247630, 5.00703186109405715450804645380, 5.49651839571519605956052754671, 5.91664450131519930752339822026, 6.98041080334226520259682781715, 7.12309892097355287467949277125, 7.61037842284467470986765016175, 7.971298553194077500011485821295, 8.985528234511624577613196419023, 9.416358635380412097278021659071, 9.604424521197358853941123539827, 10.41303416479171704547848240963, 10.91698949898015833804265695104, 11.77818916129660758820564015951
