L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s − 4·13-s + 16-s + 4·17-s − 12·23-s + 6·25-s − 4·27-s − 3·36-s + 8·39-s + 8·43-s − 2·48-s − 8·51-s + 4·52-s + 8·53-s − 24·61-s − 64-s − 4·68-s + 24·69-s − 12·75-s + 5·81-s + 12·92-s − 6·100-s − 4·101-s − 28·103-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 0.970·17-s − 2.50·23-s + 6/5·25-s − 0.769·27-s − 1/2·36-s + 1.28·39-s + 1.21·43-s − 0.288·48-s − 1.12·51-s + 0.554·52-s + 1.09·53-s − 3.07·61-s − 1/8·64-s − 0.485·68-s + 2.88·69-s − 1.38·75-s + 5/9·81-s + 1.25·92-s − 3/5·100-s − 0.398·101-s − 2.75·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1531786316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1531786316\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.998310390228690436984839026373, −8.122355185600567942613201423498, −7.930285637752151492286165146755, −7.45266808206281926041915776106, −7.39015759565747129628504855459, −6.73034273168710720939630488159, −6.42387833340555880189861123424, −5.92106320490358142633258051317, −5.78026219505288544744416246735, −5.33462430195331654459716392519, −4.89960585548359665993483001124, −4.62996282583798631118008252949, −4.13427384757802814094378207669, −3.84849602046665846154564978220, −3.31645978133578997846495246499, −2.57569096812133945889563895155, −2.36445560711683178088318063611, −1.43978161865337530302982254744, −1.14569926967265021163286808671, −0.13835932908597071675205284041,
0.13835932908597071675205284041, 1.14569926967265021163286808671, 1.43978161865337530302982254744, 2.36445560711683178088318063611, 2.57569096812133945889563895155, 3.31645978133578997846495246499, 3.84849602046665846154564978220, 4.13427384757802814094378207669, 4.62996282583798631118008252949, 4.89960585548359665993483001124, 5.33462430195331654459716392519, 5.78026219505288544744416246735, 5.92106320490358142633258051317, 6.42387833340555880189861123424, 6.73034273168710720939630488159, 7.39015759565747129628504855459, 7.45266808206281926041915776106, 7.930285637752151492286165146755, 8.122355185600567942613201423498, 8.998310390228690436984839026373