L(s) = 1 | − 2·3-s + 3·9-s − 2·13-s + 12·17-s − 2·25-s − 4·27-s − 12·29-s + 4·39-s + 16·43-s + 14·49-s − 24·51-s + 12·53-s + 20·61-s + 4·75-s − 16·79-s + 5·81-s + 24·87-s + 12·101-s − 8·103-s − 24·107-s − 36·113-s − 6·117-s + 10·121-s + 127-s − 32·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 0.554·13-s + 2.91·17-s − 2/5·25-s − 0.769·27-s − 2.22·29-s + 0.640·39-s + 2.43·43-s + 2·49-s − 3.36·51-s + 1.64·53-s + 2.56·61-s + 0.461·75-s − 1.80·79-s + 5/9·81-s + 2.57·87-s + 1.19·101-s − 0.788·103-s − 2.32·107-s − 3.38·113-s − 0.554·117-s + 0.909·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8457616133\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8457616133\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−13.10698815434128974986256965818, −12.47777993442475047537149674355, −12.09610087274707295390961533868, −11.88890078798112065973412722504, −11.20432290249664774404681518217, −10.71055167168886695063167322870, −10.21490113531554331368334859418, −9.716465742809212230492610989670, −9.386613936137468690359682807859, −8.521078990654490500488221615857, −7.70741301796249880192626477598, −7.40068433771956951407573032606, −6.97033036257582538033588600299, −5.77786494731870035702275547812, −5.66781894923398035828050631991, −5.30027144236001040420495900093, −4.11773132345347272338645197902, −3.70099785460390951534581930114, −2.45404452071008238939658620604, −1.05917143047944965551776844665,
1.05917143047944965551776844665, 2.45404452071008238939658620604, 3.70099785460390951534581930114, 4.11773132345347272338645197902, 5.30027144236001040420495900093, 5.66781894923398035828050631991, 5.77786494731870035702275547812, 6.97033036257582538033588600299, 7.40068433771956951407573032606, 7.70741301796249880192626477598, 8.521078990654490500488221615857, 9.386613936137468690359682807859, 9.716465742809212230492610989670, 10.21490113531554331368334859418, 10.71055167168886695063167322870, 11.20432290249664774404681518217, 11.88890078798112065973412722504, 12.09610087274707295390961533868, 12.47777993442475047537149674355, 13.10698815434128974986256965818