| L(s) = 1 | + 4·7-s + 9-s − 2·11-s − 12·23-s + 25-s − 12·29-s − 12·37-s + 8·43-s + 9·49-s + 8·53-s + 4·63-s − 4·67-s + 8·71-s − 8·77-s − 24·79-s − 8·81-s − 2·99-s + 4·107-s + 20·109-s + 4·113-s − 15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1/3·9-s − 0.603·11-s − 2.50·23-s + 1/5·25-s − 2.22·29-s − 1.97·37-s + 1.21·43-s + 9/7·49-s + 1.09·53-s + 0.503·63-s − 0.488·67-s + 0.949·71-s − 0.911·77-s − 2.70·79-s − 8/9·81-s − 0.201·99-s + 0.386·107-s + 1.91·109-s + 0.376·113-s − 1.36·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
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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
−8.141314443476488750561464638506, −7.68662865447357774737387377596, −7.30704441257482355077664395109, −7.11343750652475199097066346217, −6.18505589664284437344399511760, −5.71888980243021389420139118413, −5.49839351571765223735337298996, −4.89353355926751958044206949536, −4.33941336255501635028620521615, −3.94033589571250730128616378868, −3.42381378717739577121602534689, −2.37840522708261134453361444384, −1.99218422670630439404657027127, −1.41737107462760262686268260282, 0,
1.41737107462760262686268260282, 1.99218422670630439404657027127, 2.37840522708261134453361444384, 3.42381378717739577121602534689, 3.94033589571250730128616378868, 4.33941336255501635028620521615, 4.89353355926751958044206949536, 5.49839351571765223735337298996, 5.71888980243021389420139118413, 6.18505589664284437344399511760, 7.11343750652475199097066346217, 7.30704441257482355077664395109, 7.68662865447357774737387377596, 8.141314443476488750561464638506
