| L(s) = 1 | − 7-s + 2·9-s − 4·11-s + 4·23-s − 2·25-s − 8·29-s + 12·43-s + 49-s − 12·53-s − 2·63-s + 16·67-s + 4·77-s − 5·81-s − 8·99-s − 16·107-s − 8·109-s + 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 2/3·9-s − 1.20·11-s + 0.834·23-s − 2/5·25-s − 1.48·29-s + 1.82·43-s + 1/7·49-s − 1.64·53-s − 0.251·63-s + 1.95·67-s + 0.455·77-s − 5/9·81-s − 0.804·99-s − 1.54·107-s − 0.766·109-s + 3.38·113-s − 0.545·121-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 − 0.315·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
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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
−7.68663567197086182482754194500, −7.36763732696574661329900540896, −6.94432500867446569449269417611, −6.45634035097621395146590659520, −5.89601244396650050019050647414, −5.58889903946499676982938818400, −5.02899052144922273857239056481, −4.68885446617774277476054614048, −4.00529334834118058950150599189, −3.65010961868977551155237187952, −2.99530969852520525959172978235, −2.47382930354567359652258316725, −1.90127769999329170793110290770, −1.04217714896093662941733757814, 0,
1.04217714896093662941733757814, 1.90127769999329170793110290770, 2.47382930354567359652258316725, 2.99530969852520525959172978235, 3.65010961868977551155237187952, 4.00529334834118058950150599189, 4.68885446617774277476054614048, 5.02899052144922273857239056481, 5.58889903946499676982938818400, 5.89601244396650050019050647414, 6.45634035097621395146590659520, 6.94432500867446569449269417611, 7.36763732696574661329900540896, 7.68663567197086182482754194500
