| L(s) = 1 | + 2-s − 3·3-s + 5-s − 3·6-s − 7-s − 8-s + 6·9-s + 10-s − 3·11-s + 4·13-s − 14-s − 3·15-s − 16-s + 6·17-s + 6·18-s − 14·19-s + 3·21-s − 3·22-s + 6·23-s + 3·24-s + 4·26-s − 9·27-s + 6·29-s − 3·30-s + 10·31-s + 9·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 0.447·5-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.904·11-s + 1.10·13-s − 0.267·14-s − 0.774·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 3.21·19-s + 0.654·21-s − 0.639·22-s + 1.25·23-s + 0.612·24-s + 0.784·26-s − 1.73·27-s + 1.11·29-s − 0.547·30-s + 1.79·31-s + 1.56·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231170499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231170499\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.69807178842631840647451144202, −10.56760135807349868326809112196, −10.13741094357174451878621281532, −9.829016613268438883221680717459, −9.120230554505570598888224331785, −8.628840764224520108233239347229, −7.984796346894655846101931496928, −7.909870642544675970084118990727, −6.77331916281562260774049178943, −6.55663842289685601167195634305, −6.15687998807320618404880839805, −6.04943749075796288599573865604, −5.11816319664591450367153839155, −5.11275081329142052909592205404, −4.26150984347762003833465096269, −4.17347104829828329042001660538, −3.08904956914175562871354698319, −2.63094255487437300804473523083, −1.52114380416761937174305248058, −0.63835042020437344778976500790,
0.63835042020437344778976500790, 1.52114380416761937174305248058, 2.63094255487437300804473523083, 3.08904956914175562871354698319, 4.17347104829828329042001660538, 4.26150984347762003833465096269, 5.11275081329142052909592205404, 5.11816319664591450367153839155, 6.04943749075796288599573865604, 6.15687998807320618404880839805, 6.55663842289685601167195634305, 6.77331916281562260774049178943, 7.909870642544675970084118990727, 7.984796346894655846101931496928, 8.628840764224520108233239347229, 9.120230554505570598888224331785, 9.829016613268438883221680717459, 10.13741094357174451878621281532, 10.56760135807349868326809112196, 10.69807178842631840647451144202
