| L(s) = 1 | − 28·11-s − 38·19-s + 28·29-s + 266·31-s − 168·41-s + 661·49-s − 388·59-s − 34·61-s − 1.65e3·71-s + 1.10e3·79-s − 2.20e3·89-s − 1.10e3·101-s + 3.68e3·109-s − 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.767·11-s − 0.458·19-s + 0.179·29-s + 1.54·31-s − 0.639·41-s + 1.92·49-s − 0.856·59-s − 0.0713·61-s − 2.76·71-s + 1.57·79-s − 2.62·89-s − 1.08·101-s + 3.23·109-s − 1.55·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.013779520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013779520\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 661 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7710 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 133 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34742 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 131125 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 39546 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 89818 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 194 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 175117 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 828 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 453134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 552 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1123410 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1104 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1118065 T^{2} + p^{6} T^{4} \) |
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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
−9.082133900033455591715144742662, −8.617190674675728636225261170691, −8.433450826693909415053468609062, −7.88061089006616393460637474878, −7.62286755274128728592367266603, −7.13212757852434674979840553204, −6.78660211770914979109623115383, −6.23860887190096125143912679361, −5.97713656241324522250851016807, −5.45846272535351158481225122956, −5.07628384929594705147211156638, −4.48464602346167380237537251035, −4.32212075797690691869831929344, −3.65745635718767971466762032472, −3.12784551227140321830914791744, −2.59410291728196429053821993872, −2.34057615370752309759200190601, −1.50150977452343665718071761984, −1.03859485927879428294446752631, −0.23136645742809323516223790514,
0.23136645742809323516223790514, 1.03859485927879428294446752631, 1.50150977452343665718071761984, 2.34057615370752309759200190601, 2.59410291728196429053821993872, 3.12784551227140321830914791744, 3.65745635718767971466762032472, 4.32212075797690691869831929344, 4.48464602346167380237537251035, 5.07628384929594705147211156638, 5.45846272535351158481225122956, 5.97713656241324522250851016807, 6.23860887190096125143912679361, 6.78660211770914979109623115383, 7.13212757852434674979840553204, 7.62286755274128728592367266603, 7.88061089006616393460637474878, 8.433450826693909415053468609062, 8.617190674675728636225261170691, 9.082133900033455591715144742662
