| L(s) = 1 | − 3-s − 4-s − 2·9-s − 5·11-s + 12-s − 3·16-s − 13·23-s + 5·27-s − 5·31-s + 5·33-s + 2·36-s − 5·37-s + 5·44-s − 11·47-s + 3·48-s + 11·49-s − 12·53-s − 20·59-s + 7·64-s + 13·69-s − 10·71-s + 81-s − 15·89-s + 13·92-s + 5·93-s + 5·97-s + 10·99-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 2/3·9-s − 1.50·11-s + 0.288·12-s − 3/4·16-s − 2.71·23-s + 0.962·27-s − 0.898·31-s + 0.870·33-s + 1/3·36-s − 0.821·37-s + 0.753·44-s − 1.60·47-s + 0.433·48-s + 11/7·49-s − 1.64·53-s − 2.60·59-s + 7/8·64-s + 1.56·69-s − 1.18·71-s + 1/9·81-s − 1.58·89-s + 1.35·92-s + 0.518·93-s + 0.507·97-s + 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.984631823591704409584288553009, −7.46844877994892620924386881920, −7.09698062395090384737132598561, −6.25917433233698885877794056999, −6.06418929997933995351356743814, −5.65991308891887299221535977745, −5.04854762245207400347329510971, −4.74734482148070973567586964597, −4.21520932563468592372816179713, −3.52991956046757587806494162300, −2.96423227830079656053098250026, −2.28019862371409758309199552547, −1.68277288485814467930946781424, 0, 0,
1.68277288485814467930946781424, 2.28019862371409758309199552547, 2.96423227830079656053098250026, 3.52991956046757587806494162300, 4.21520932563468592372816179713, 4.74734482148070973567586964597, 5.04854762245207400347329510971, 5.65991308891887299221535977745, 6.06418929997933995351356743814, 6.25917433233698885877794056999, 7.09698062395090384737132598561, 7.46844877994892620924386881920, 7.984631823591704409584288553009
