| L(s) = 1 | − 136·7-s + 32·13-s − 416·19-s + 3.30e3·31-s + 884·37-s − 2.32e3·43-s + 9.07e3·49-s − 7.82e3·61-s − 1.27e4·67-s + 4.44e3·73-s − 1.41e4·79-s − 4.35e3·91-s − 8.70e3·97-s + 1.47e4·103-s − 2.88e4·109-s + 5.95e3·121-s + 127-s + 131-s + 5.65e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.63e4·169-s + ⋯ |
| L(s) = 1 | − 2.77·7-s + 0.189·13-s − 1.15·19-s + 3.43·31-s + 0.645·37-s − 1.25·43-s + 3.77·49-s − 2.10·61-s − 2.84·67-s + 0.834·73-s − 2.26·79-s − 0.525·91-s − 0.925·97-s + 1.39·103-s − 2.42·109-s + 0.406·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 3.19·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 1.97·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.009178207789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.009178207789\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + 68 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 5954 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 79360 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 208 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 536354 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 993200 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 1652 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 442 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5405120 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1160 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9549410 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9620912 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12943970 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3910 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6392 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7689890 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2224 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7060 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 59434754 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 54274304 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4352 T + p^{4} T^{2} )^{2} \) |
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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
−9.796918664980279672215945729475, −9.422568541350950122935682214027, −9.035973903942513467723913447451, −8.421289646705206709472313216606, −8.299040483160485230032442442654, −7.53672955626185182445345184035, −7.10401823441316658599004909486, −6.44383551880429247370706486333, −6.43549876436679640661422490361, −6.12191493461790665842495184788, −5.63300250413059632914994934359, −4.65573504649488704945291843957, −4.48213242624024150022080115645, −3.83872599729642944922108089431, −3.24974359063836895360898509803, −2.75084735975970116065708391691, −2.72179269169663810343093217466, −1.59731227133721942137030088334, −0.889339746336752499515486252884, −0.02561304880034246965318384004,
0.02561304880034246965318384004, 0.889339746336752499515486252884, 1.59731227133721942137030088334, 2.72179269169663810343093217466, 2.75084735975970116065708391691, 3.24974359063836895360898509803, 3.83872599729642944922108089431, 4.48213242624024150022080115645, 4.65573504649488704945291843957, 5.63300250413059632914994934359, 6.12191493461790665842495184788, 6.43549876436679640661422490361, 6.44383551880429247370706486333, 7.10401823441316658599004909486, 7.53672955626185182445345184035, 8.299040483160485230032442442654, 8.421289646705206709472313216606, 9.035973903942513467723913447451, 9.422568541350950122935682214027, 9.796918664980279672215945729475
