| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 3·7-s + 8-s − 11-s − 2·13-s − 3·14-s − 16-s + 6·17-s − 6·19-s − 6·21-s − 22-s + 5·23-s + 2·24-s + 2·25-s − 2·26-s − 2·27-s + 2·29-s − 3·31-s − 6·32-s − 2·33-s + 6·34-s − 3·37-s − 6·38-s − 4·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.816·6-s − 1.13·7-s + 0.353·8-s − 0.301·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s + 1.45·17-s − 1.37·19-s − 1.30·21-s − 0.213·22-s + 1.04·23-s + 0.408·24-s + 2/5·25-s − 0.392·26-s − 0.384·27-s + 0.371·29-s − 0.538·31-s − 1.06·32-s − 0.348·33-s + 1.02·34-s − 0.493·37-s − 0.973·38-s − 0.640·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7831 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7831 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526667399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526667399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 21 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 55 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 197 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 162 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−16.7473493471, −16.4200547175, −15.8434468443, −15.0970471591, −14.7847597830, −14.4772648201, −13.8228849406, −13.3537432347, −13.1259605375, −12.3728783405, −12.1621933930, −11.2021686118, −10.4278952958, −10.1958954359, −9.33957173169, −8.86177755711, −8.42392919163, −7.55699998798, −7.05413997648, −6.31417875291, −5.42480979866, −4.76943118095, −3.75511202152, −3.17530913794, −2.35388154380,
2.35388154380, 3.17530913794, 3.75511202152, 4.76943118095, 5.42480979866, 6.31417875291, 7.05413997648, 7.55699998798, 8.42392919163, 8.86177755711, 9.33957173169, 10.1958954359, 10.4278952958, 11.2021686118, 12.1621933930, 12.3728783405, 13.1259605375, 13.3537432347, 13.8228849406, 14.4772648201, 14.7847597830, 15.0970471591, 15.8434468443, 16.4200547175, 16.7473493471
