| L(s) = 1 | − 4-s − 6·13-s + 16-s + 4·17-s − 8·23-s + 6·25-s + 20·29-s + 8·43-s + 10·49-s + 6·52-s + 12·53-s + 4·61-s − 64-s − 4·68-s + 8·92-s − 6·100-s − 4·101-s − 32·103-s − 16·107-s − 28·113-s − 20·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.66·23-s + 6/5·25-s + 3.71·29-s + 1.21·43-s + 10/7·49-s + 0.832·52-s + 1.64·53-s + 0.512·61-s − 1/8·64-s − 0.485·68-s + 0.834·92-s − 3/5·100-s − 0.398·101-s − 3.15·103-s − 1.54·107-s − 2.63·113-s − 1.85·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157017170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157017170\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−12.38284341145745979941739153293, −12.10378650439117253917267311756, −11.74274292373543036929049707771, −10.76402192936562144044738310590, −10.34969921396913105751686373402, −10.08091823126173197286688880056, −9.634127354878424262414246398205, −9.021502634551145273383072217145, −8.401786004937940332109564926983, −8.100307463457145770790697528030, −7.48072179151526523019388320071, −6.88883468919043707757735434290, −6.42094053815453976712836435437, −5.54880825876954515067440994214, −5.20118278645230288588682717410, −4.42018691647936959828189349714, −4.07498965048745238112658744397, −2.88002558478575492595584477474, −2.49805887825902958986511875731, −0.942953026571404839167082239747,
0.942953026571404839167082239747, 2.49805887825902958986511875731, 2.88002558478575492595584477474, 4.07498965048745238112658744397, 4.42018691647936959828189349714, 5.20118278645230288588682717410, 5.54880825876954515067440994214, 6.42094053815453976712836435437, 6.88883468919043707757735434290, 7.48072179151526523019388320071, 8.100307463457145770790697528030, 8.401786004937940332109564926983, 9.021502634551145273383072217145, 9.634127354878424262414246398205, 10.08091823126173197286688880056, 10.34969921396913105751686373402, 10.76402192936562144044738310590, 11.74274292373543036929049707771, 12.10378650439117253917267311756, 12.38284341145745979941739153293
