| L(s) = 1 | − 2·2-s − 5-s + 2·7-s + 4·8-s − 5·9-s + 2·10-s − 6·11-s + 4·13-s − 4·14-s − 4·16-s − 4·17-s + 10·18-s + 2·19-s + 12·22-s − 12·23-s + 25-s − 8·26-s − 2·29-s − 2·31-s + 8·34-s − 2·35-s − 4·38-s − 4·40-s − 10·41-s − 6·43-s + 5·45-s + 24·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.447·5-s + 0.755·7-s + 1.41·8-s − 5/3·9-s + 0.632·10-s − 1.80·11-s + 1.10·13-s − 1.06·14-s − 16-s − 0.970·17-s + 2.35·18-s + 0.458·19-s + 2.55·22-s − 2.50·23-s + 1/5·25-s − 1.56·26-s − 0.371·29-s − 0.359·31-s + 1.37·34-s − 0.338·35-s − 0.648·38-s − 0.632·40-s − 1.56·41-s − 0.914·43-s + 0.745·45-s + 3.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−17.7580669006, −17.1435628879, −16.5956405490, −16.1943353647, −15.4709418010, −15.2725263569, −14.1437362255, −14.0962101507, −13.3483260723, −13.0993512532, −11.8074747577, −11.7492759155, −10.9940749680, −10.2860419224, −10.2617561341, −9.09817427128, −8.67949885325, −8.27407602891, −7.98094729632, −7.30330020618, −5.99848698023, −5.47907564041, −4.63171310479, −3.61689003046, −2.25267400471, 0,
2.25267400471, 3.61689003046, 4.63171310479, 5.47907564041, 5.99848698023, 7.30330020618, 7.98094729632, 8.27407602891, 8.67949885325, 9.09817427128, 10.2617561341, 10.2860419224, 10.9940749680, 11.7492759155, 11.8074747577, 13.0993512532, 13.3483260723, 14.0962101507, 14.1437362255, 15.2725263569, 15.4709418010, 16.1943353647, 16.5956405490, 17.1435628879, 17.7580669006
