| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s + 12-s + 3·15-s + 16-s − 18-s − 10·19-s + 3·20-s − 4·23-s − 24-s + 2·25-s + 27-s − 7·29-s − 3·30-s − 32-s + 36-s + 10·38-s − 3·40-s − 2·43-s + 3·45-s + 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.235·18-s − 2.29·19-s + 0.670·20-s − 0.834·23-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 1.29·29-s − 0.547·30-s − 0.176·32-s + 1/6·36-s + 1.62·38-s − 0.474·40-s − 0.304·43-s + 0.447·45-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−8.355570118234955840150658160221, −7.904166027792475331391748882302, −7.59485095469953497537541609413, −6.80783030247249511213699205531, −6.48454696538758804016091320147, −6.11604317911845432279747962322, −5.70549381705263656275552912828, −4.98993511836650092967717183574, −4.47107962271686706319330087016, −3.79879224866438289559029577033, −3.22234973889576120764839865408, −2.39785843394366783207174907850, −1.94592661427331401337015483963, −1.60673847639186768733616673958, 0,
1.60673847639186768733616673958, 1.94592661427331401337015483963, 2.39785843394366783207174907850, 3.22234973889576120764839865408, 3.79879224866438289559029577033, 4.47107962271686706319330087016, 4.98993511836650092967717183574, 5.70549381705263656275552912828, 6.11604317911845432279747962322, 6.48454696538758804016091320147, 6.80783030247249511213699205531, 7.59485095469953497537541609413, 7.904166027792475331391748882302, 8.355570118234955840150658160221
