| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 18-s + 4·19-s + 20-s + 24-s − 2·25-s − 27-s + 5·29-s + 30-s − 32-s + 36-s − 4·38-s − 40-s − 4·43-s + 45-s + 20·47-s − 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.204·24-s − 2/5·25-s − 0.192·27-s + 0.928·29-s + 0.182·30-s − 0.176·32-s + 1/6·36-s − 0.648·38-s − 0.158·40-s − 0.609·43-s + 0.149·45-s + 2.91·47-s − 0.144·48-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
| \(L(1)\) |
\(\approx\) |
\(1.222142256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222142256\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.537115304716422550180749049136, −8.148215759883696625761538787072, −7.40369966731023467746884540309, −7.29659641648074087179168350226, −6.79520491120165194072926447039, −6.11157023291586646171362059203, −5.89246278809542126832461257799, −5.31940454596833772548449257157, −4.91906774659403620128694795709, −4.10985732909137679075427303871, −3.70795849171544437978395948028, −2.77947313614275915127380172608, −2.36483974362438558904110582850, −1.44693951614241138604182923191, −0.72559796503309483236519530203,
0.72559796503309483236519530203, 1.44693951614241138604182923191, 2.36483974362438558904110582850, 2.77947313614275915127380172608, 3.70795849171544437978395948028, 4.10985732909137679075427303871, 4.91906774659403620128694795709, 5.31940454596833772548449257157, 5.89246278809542126832461257799, 6.11157023291586646171362059203, 6.79520491120165194072926447039, 7.29659641648074087179168350226, 7.40369966731023467746884540309, 8.148215759883696625761538787072, 8.537115304716422550180749049136
