| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 9-s − 2·11-s + 2·12-s − 6·13-s − 16-s − 18-s + 2·22-s − 8·23-s − 6·24-s − 6·25-s + 6·26-s + 4·27-s − 5·32-s + 4·33-s − 36-s − 4·37-s + 12·39-s + 2·44-s + 8·46-s − 11·47-s + 2·48-s − 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.66·13-s − 1/4·16-s − 0.235·18-s + 0.426·22-s − 1.66·23-s − 1.22·24-s − 6/5·25-s + 1.17·26-s + 0.769·27-s − 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.657·37-s + 1.92·39-s + 0.301·44-s + 1.17·46-s − 1.60·47-s + 0.288·48-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.70688777284638673979565451134, −11.03739671080096490885210313851, −10.21452845382174683769874300557, −10.03117785392807924121407255636, −9.615382857462666609097306020184, −8.642154117411816364534592559742, −8.082628173026923360397743359668, −7.53156244480756256987795816140, −6.83984285015980646588858775247, −5.95849882512141861238903031446, −5.24441889789184721049990949223, −4.79790534272746541152460548441, −3.82226744501162954417248868453, −2.19723254370844623474510707354, 0,
2.19723254370844623474510707354, 3.82226744501162954417248868453, 4.79790534272746541152460548441, 5.24441889789184721049990949223, 5.95849882512141861238903031446, 6.83984285015980646588858775247, 7.53156244480756256987795816140, 8.082628173026923360397743359668, 8.642154117411816364534592559742, 9.615382857462666609097306020184, 10.03117785392807924121407255636, 10.21452845382174683769874300557, 11.03739671080096490885210313851, 11.70688777284638673979565451134
