| L(s) = 1 | + 2·3-s − 2·5-s − 4·7-s + 9-s − 4·15-s + 8·17-s − 8·21-s − 25-s − 4·27-s + 8·35-s − 16·37-s + 8·41-s + 8·43-s − 2·45-s + 8·47-s + 9·49-s + 16·51-s − 4·63-s − 8·67-s − 2·75-s + 16·79-s − 11·81-s − 12·83-s − 16·85-s − 8·89-s + 16·101-s + 16·105-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.03·15-s + 1.94·17-s − 1.74·21-s − 1/5·25-s − 0.769·27-s + 1.35·35-s − 2.63·37-s + 1.24·41-s + 1.21·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 2.24·51-s − 0.503·63-s − 0.977·67-s − 0.230·75-s + 1.80·79-s − 1.22·81-s − 1.31·83-s − 1.73·85-s − 0.847·89-s + 1.59·101-s + 1.56·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622798493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622798493\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−8.249132901246965294194546216109, −7.81121777124047910458865164762, −7.56939034097939823411596630436, −7.13394608071504157568990120893, −6.69282662507338911319868152739, −5.96771731363836041408875304159, −5.68004045476398576609004806079, −5.18568247712385326671148326141, −4.24124226993375610448374416528, −3.86283044993670820068383948685, −3.46582162948881971467644182437, −3.04785850968902782895844059133, −2.60939931609178692198772762353, −1.69826006157989031969230383507, −0.58165905632287903870280513988,
0.58165905632287903870280513988, 1.69826006157989031969230383507, 2.60939931609178692198772762353, 3.04785850968902782895844059133, 3.46582162948881971467644182437, 3.86283044993670820068383948685, 4.24124226993375610448374416528, 5.18568247712385326671148326141, 5.68004045476398576609004806079, 5.96771731363836041408875304159, 6.69282662507338911319868152739, 7.13394608071504157568990120893, 7.56939034097939823411596630436, 7.81121777124047910458865164762, 8.249132901246965294194546216109
