L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s + 4·17-s − 8·23-s − 4·27-s − 20·29-s − 3·36-s − 12·39-s − 8·43-s − 2·48-s + 10·49-s − 8·51-s − 6·52-s + 12·53-s + 4·61-s − 64-s − 4·68-s + 16·69-s + 5·81-s + 40·87-s + 8·92-s + 4·101-s + 32·103-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.66·23-s − 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s − 1.21·43-s − 0.288·48-s + 10/7·49-s − 1.12·51-s − 0.832·52-s + 1.64·53-s + 0.512·61-s − 1/8·64-s − 0.485·68-s + 1.92·69-s + 5/9·81-s + 4.28·87-s + 0.834·92-s + 0.398·101-s + 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8554223163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8554223163\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.233566324728389761112545598671, −8.956616267703219010374133862162, −8.872546661394092685504524929219, −7.986726649373977717944633570727, −7.82365178246178017393697956803, −7.58302181044842264236368034285, −6.86151622954446610530335012247, −6.63300166738852675882914303650, −5.95411416925638363048231088412, −5.80448407719749045938899759697, −5.38021803883807541705352987189, −5.28563608774151097138914480545, −4.36036393482824586285885686025, −4.00468394201600375013425422533, −3.67416244534569336265678746531, −3.42392193044851321230971858424, −2.32606893234971786995695680463, −1.76108113140118599472720261700, −1.23703861344529045297577011933, −0.39983921122863822593076142449,
0.39983921122863822593076142449, 1.23703861344529045297577011933, 1.76108113140118599472720261700, 2.32606893234971786995695680463, 3.42392193044851321230971858424, 3.67416244534569336265678746531, 4.00468394201600375013425422533, 4.36036393482824586285885686025, 5.28563608774151097138914480545, 5.38021803883807541705352987189, 5.80448407719749045938899759697, 5.95411416925638363048231088412, 6.63300166738852675882914303650, 6.86151622954446610530335012247, 7.58302181044842264236368034285, 7.82365178246178017393697956803, 7.986726649373977717944633570727, 8.872546661394092685504524929219, 8.956616267703219010374133862162, 9.233566324728389761112545598671