| L(s) = 1 | − 190·9-s + 16·11-s + 5.39e3·19-s + 6.50e3·29-s + 9.57e3·31-s + 2.67e4·41-s − 2.40e3·49-s − 6.94e4·59-s − 2.06e3·61-s + 1.25e5·71-s − 2.28e4·79-s − 2.29e4·81-s − 3.94e4·89-s − 3.04e3·99-s + 9.18e4·101-s + 2.92e5·109-s − 3.21e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.74e5·169-s + ⋯ |
| L(s) = 1 | − 0.781·9-s + 0.0398·11-s + 3.42·19-s + 1.43·29-s + 1.78·31-s + 2.48·41-s − 1/7·49-s − 2.59·59-s − 0.0710·61-s + 2.95·71-s − 0.411·79-s − 0.388·81-s − 0.527·89-s − 0.0311·99-s + 0.895·101-s + 2.36·109-s − 1.99·121-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 + 0.739·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.786968494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.786968494\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 190 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 274730 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2079810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 142 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1690350 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3254 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4788 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 5206 p^{2} T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 13350 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293155702 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 457221070 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 664518886 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 34702 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1032 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2598078550 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 62720 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3787949710 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11400 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 35444478 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 19722 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16883568670 T^{2} + p^{10} T^{4} \) |
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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
−9.978618610704469812879841866272, −9.508189959917088312837977440987, −9.041343905110711834432064447421, −8.728276621716262929981221106362, −7.943532404469563784487236275034, −7.81210433870065799153423810838, −7.48228842517897473000182117728, −6.74447270860215779604596003012, −6.39585942779594544159095719289, −5.78185358935566345258333302199, −5.53480715811095869125605992734, −4.82973845970495368590942040896, −4.63685170583588578708517041723, −3.82848343290785251648986693058, −3.10962668352713938753962872717, −2.96721600251874651710041593643, −2.42662523741341797727763572820, −1.43653113787135412477803022112, −0.877851811783822040198685488929, −0.61082719206203122898371408343,
0.61082719206203122898371408343, 0.877851811783822040198685488929, 1.43653113787135412477803022112, 2.42662523741341797727763572820, 2.96721600251874651710041593643, 3.10962668352713938753962872717, 3.82848343290785251648986693058, 4.63685170583588578708517041723, 4.82973845970495368590942040896, 5.53480715811095869125605992734, 5.78185358935566345258333302199, 6.39585942779594544159095719289, 6.74447270860215779604596003012, 7.48228842517897473000182117728, 7.81210433870065799153423810838, 7.943532404469563784487236275034, 8.728276621716262929981221106362, 9.041343905110711834432064447421, 9.508189959917088312837977440987, 9.978618610704469812879841866272
