| L(s) = 1 | − 10·19-s − 12·29-s + 10·31-s − 24·41-s + 10·49-s + 12·59-s − 14·61-s − 24·71-s + 2·79-s − 24·89-s − 24·101-s + 14·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2.29·19-s − 2.22·29-s + 1.79·31-s − 3.74·41-s + 10/7·49-s + 1.56·59-s − 1.79·61-s − 2.84·71-s + 0.225·79-s − 2.54·89-s − 2.38·101-s + 1.34·109-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3967471586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3967471586\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.965139625393844551446875413928, −8.533760035414045311405967542191, −8.507701003224320979849503734115, −7.74900217606276847822930402353, −7.74070404352300971470179119877, −6.90644493267818336761274269203, −6.83193214428123948603486036627, −6.43073832395284345494871849228, −5.99494501538510585038114112460, −5.43131183881456093268905833899, −5.32564962376082343222952214163, −4.47530553601358451172067260392, −4.43171207048348375439391797315, −3.81462926068523236908028476419, −3.51657820152373293147002787362, −2.69345992915804009536340163311, −2.53406835853089939199932237955, −1.61374302097108356936335356159, −1.54933301357522248129436025119, −0.19522241933984479245536358872,
0.19522241933984479245536358872, 1.54933301357522248129436025119, 1.61374302097108356936335356159, 2.53406835853089939199932237955, 2.69345992915804009536340163311, 3.51657820152373293147002787362, 3.81462926068523236908028476419, 4.43171207048348375439391797315, 4.47530553601358451172067260392, 5.32564962376082343222952214163, 5.43131183881456093268905833899, 5.99494501538510585038114112460, 6.43073832395284345494871849228, 6.83193214428123948603486036627, 6.90644493267818336761274269203, 7.74070404352300971470179119877, 7.74900217606276847822930402353, 8.507701003224320979849503734115, 8.533760035414045311405967542191, 8.965139625393844551446875413928
