| L(s) = 1 | + 2·4-s − 2·11-s − 6·13-s − 14·17-s + 2·23-s − 4·25-s − 6·31-s − 10·41-s − 4·44-s − 20·47-s − 8·49-s − 12·52-s − 2·53-s − 20·59-s − 8·64-s + 18·67-s − 28·68-s − 12·79-s − 9·81-s + 2·83-s + 4·92-s + 22·97-s − 8·100-s − 10·101-s + 34·103-s + 28·107-s + 26·109-s + ⋯ |
| L(s) = 1 | + 4-s − 0.603·11-s − 1.66·13-s − 3.39·17-s + 0.417·23-s − 4/5·25-s − 1.07·31-s − 1.56·41-s − 0.603·44-s − 2.91·47-s − 8/7·49-s − 1.66·52-s − 0.274·53-s − 2.60·59-s − 64-s + 2.19·67-s − 3.39·68-s − 1.35·79-s − 81-s + 0.219·83-s + 0.417·92-s + 2.23·97-s − 4/5·100-s − 0.995·101-s + 3.35·103-s + 2.70·107-s + 2.49·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 43 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−9.073381058447805601256365309005, −8.649842777033441119852664742335, −8.337089079911450708883195215285, −7.60584991633589573479032289791, −7.53559212230323752887327208127, −7.04407308951021410917296057822, −6.67863814172410917138263516897, −6.25002369939981202080222924059, −6.21377957399014347954774776352, −5.19349748322025490914723346512, −4.91963650750806191459270266519, −4.66217848583934006532747445107, −4.23292310430270266036539560339, −3.21098522528394051622634132865, −3.19392624763796702691461080328, −2.31122467508602671580007165175, −1.92387577974938266647656403220, −1.88726173175353408736274236765, 0, 0,
1.88726173175353408736274236765, 1.92387577974938266647656403220, 2.31122467508602671580007165175, 3.19392624763796702691461080328, 3.21098522528394051622634132865, 4.23292310430270266036539560339, 4.66217848583934006532747445107, 4.91963650750806191459270266519, 5.19349748322025490914723346512, 6.21377957399014347954774776352, 6.25002369939981202080222924059, 6.67863814172410917138263516897, 7.04407308951021410917296057822, 7.53559212230323752887327208127, 7.60584991633589573479032289791, 8.337089079911450708883195215285, 8.649842777033441119852664742335, 9.073381058447805601256365309005
