L(s) = 1 | − 2·5-s − 12·11-s − 12·19-s − 25-s − 4·29-s + 20·31-s − 4·41-s − 49-s + 24·55-s + 16·59-s − 4·61-s + 20·71-s + 8·79-s + 12·89-s + 24·95-s + 12·101-s − 4·109-s + 86·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 40·155-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 3.61·11-s − 2.75·19-s − 1/5·25-s − 0.742·29-s + 3.59·31-s − 0.624·41-s − 1/7·49-s + 3.23·55-s + 2.08·59-s − 0.512·61-s + 2.37·71-s + 0.900·79-s + 1.27·89-s + 2.46·95-s + 1.19·101-s − 0.383·109-s + 7.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 3.21·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8988936580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8988936580\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.361835118147274635800141074582, −8.019167932013457252481775158744, −7.70625668077413226833832602500, −7.69079784765016016213972563289, −6.97403708547201024351216230581, −6.63075326421363053411211455733, −6.23338218497328951643090127303, −5.99481893428776636770374903919, −5.23252694421221913589755940860, −5.21264409329943750214104591463, −4.71565485860379540356833093134, −4.46823829278461423382501114175, −3.94985970118606500879049095822, −3.57058368523390818353692304276, −2.81268314501469923363982832570, −2.75377545149421420779722922875, −2.11936001204954947106587096675, −2.05209531792832831436100079288, −0.66703873303515375556556336169, −0.38284125303428490294899336549,
0.38284125303428490294899336549, 0.66703873303515375556556336169, 2.05209531792832831436100079288, 2.11936001204954947106587096675, 2.75377545149421420779722922875, 2.81268314501469923363982832570, 3.57058368523390818353692304276, 3.94985970118606500879049095822, 4.46823829278461423382501114175, 4.71565485860379540356833093134, 5.21264409329943750214104591463, 5.23252694421221913589755940860, 5.99481893428776636770374903919, 6.23338218497328951643090127303, 6.63075326421363053411211455733, 6.97403708547201024351216230581, 7.69079784765016016213972563289, 7.70625668077413226833832602500, 8.019167932013457252481775158744, 8.361835118147274635800141074582