| L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s + 3·11-s − 16-s − 2·18-s − 3·22-s + 8·23-s − 6·25-s − 12·29-s − 5·32-s − 2·36-s + 4·37-s − 3·44-s − 8·46-s − 7·49-s + 6·50-s − 12·53-s + 12·58-s + 7·64-s − 8·67-s + 6·72-s − 4·74-s + 16·79-s − 5·81-s + 9·88-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 0.904·11-s − 1/4·16-s − 0.471·18-s − 0.639·22-s + 1.66·23-s − 6/5·25-s − 2.22·29-s − 0.883·32-s − 1/3·36-s + 0.657·37-s − 0.452·44-s − 1.17·46-s − 49-s + 0.848·50-s − 1.64·53-s + 1.57·58-s + 7/8·64-s − 0.977·67-s + 0.707·72-s − 0.464·74-s + 1.80·79-s − 5/9·81-s + 0.959·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5519183913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5519183913\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.57313208072039565242489501283, −11.63316742955686156406170012524, −11.16672268822472124045955896956, −10.62530427950729378876476435816, −9.765679143838034283560120231075, −9.383910769388270541310975434624, −9.060882887232174526671976429397, −8.125068841280254498307837446517, −7.56224037698736452894490709631, −6.97747003095705240814495677956, −6.07772411408113041386088890828, −5.12124979070812192521635742220, −4.31769654779329217122275139856, −3.51333942727764849161163416179, −1.62756022206303990072823312538,
1.62756022206303990072823312538, 3.51333942727764849161163416179, 4.31769654779329217122275139856, 5.12124979070812192521635742220, 6.07772411408113041386088890828, 6.97747003095705240814495677956, 7.56224037698736452894490709631, 8.125068841280254498307837446517, 9.060882887232174526671976429397, 9.383910769388270541310975434624, 9.765679143838034283560120231075, 10.62530427950729378876476435816, 11.16672268822472124045955896956, 11.63316742955686156406170012524, 12.57313208072039565242489501283
