| L(s) = 1 | − 4-s + 5·9-s + 2·11-s + 16-s − 10·19-s − 10·29-s − 6·31-s − 5·36-s + 4·41-s − 2·44-s + 5·49-s + 20·59-s + 14·61-s − 64-s + 14·71-s + 10·76-s − 20·79-s + 16·81-s + 30·89-s + 10·99-s + 4·101-s + 20·109-s + 10·116-s + 3·121-s + 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s + 0.603·11-s + 1/4·16-s − 2.29·19-s − 1.85·29-s − 1.07·31-s − 5/6·36-s + 0.624·41-s − 0.301·44-s + 5/7·49-s + 2.60·59-s + 1.79·61-s − 1/8·64-s + 1.66·71-s + 1.14·76-s − 2.25·79-s + 16/9·81-s + 3.17·89-s + 1.00·99-s + 0.398·101-s + 1.91·109-s + 0.928·116-s + 3/11·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.639320205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639320205\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−11.30507882954410098538484757380, −10.54511783684765172753483428333, −9.924985562807607904183772242187, −9.901084917741441723928456229539, −9.216441747453579076338936149054, −8.833590833937272714194167185340, −8.505081785704659628194314697391, −7.85492549886027616160729448087, −7.39209812944101491682247509674, −6.85487389309921397567274826896, −6.71538491869990097464655362134, −5.84280563667207028487462762207, −5.57241688825574351725069640204, −4.70587621601681675417897440204, −4.36770169222068280176371463352, −3.75007429883438060249741022758, −3.65203058309269662712530383262, −2.06462701055061654963223705148, −2.04599276658663454121497080206, −0.76887550019713147419647615256,
0.76887550019713147419647615256, 2.04599276658663454121497080206, 2.06462701055061654963223705148, 3.65203058309269662712530383262, 3.75007429883438060249741022758, 4.36770169222068280176371463352, 4.70587621601681675417897440204, 5.57241688825574351725069640204, 5.84280563667207028487462762207, 6.71538491869990097464655362134, 6.85487389309921397567274826896, 7.39209812944101491682247509674, 7.85492549886027616160729448087, 8.505081785704659628194314697391, 8.833590833937272714194167185340, 9.216441747453579076338936149054, 9.901084917741441723928456229539, 9.924985562807607904183772242187, 10.54511783684765172753483428333, 11.30507882954410098538484757380
