| L(s) = 1 | + 2-s + 2·4-s + 5·8-s + 4·11-s + 5·16-s + 4·22-s + 8·23-s + 5·25-s − 4·29-s + 10·32-s + 6·37-s − 24·43-s + 8·44-s + 8·46-s + 5·50-s − 10·53-s − 4·58-s + 17·64-s − 4·67-s − 32·71-s + 6·74-s − 8·79-s − 24·86-s + 20·88-s + 16·92-s + 10·100-s − 10·106-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 1.76·8-s + 1.20·11-s + 5/4·16-s + 0.852·22-s + 1.66·23-s + 25-s − 0.742·29-s + 1.76·32-s + 0.986·37-s − 3.65·43-s + 1.20·44-s + 1.17·46-s + 0.707·50-s − 1.37·53-s − 0.525·58-s + 17/8·64-s − 0.488·67-s − 3.79·71-s + 0.697·74-s − 0.900·79-s − 2.58·86-s + 2.13·88-s + 1.66·92-s + 100-s − 0.971·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.737694151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.737694151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−11.22118699307712351797105705482, −11.18803891444180594550764081904, −10.42260184288219637129287499863, −10.30471708388913224954778517263, −9.423907224831202791519044739523, −9.308040957925004218281811246323, −8.394894458611223772036190976834, −8.287968396041543380977247870036, −7.29139196053559166721759899932, −7.24377531406218404291946005590, −6.62665534373549917236332618591, −6.36579893512035772327223491545, −5.58751412495102888234703243537, −5.01970630315614486339063133602, −4.53634707812790858538234661528, −4.13031728033590283530223709279, −3.17051580992645947470498223898, −2.97309757549757489362482940577, −1.72308377574922347240824843108, −1.36244013126385186572929554582,
1.36244013126385186572929554582, 1.72308377574922347240824843108, 2.97309757549757489362482940577, 3.17051580992645947470498223898, 4.13031728033590283530223709279, 4.53634707812790858538234661528, 5.01970630315614486339063133602, 5.58751412495102888234703243537, 6.36579893512035772327223491545, 6.62665534373549917236332618591, 7.24377531406218404291946005590, 7.29139196053559166721759899932, 8.287968396041543380977247870036, 8.394894458611223772036190976834, 9.308040957925004218281811246323, 9.423907224831202791519044739523, 10.30471708388913224954778517263, 10.42260184288219637129287499863, 11.18803891444180594550764081904, 11.22118699307712351797105705482
