| L(s) = 1 | + 2-s − 4-s − 3·8-s + 9-s − 8·13-s − 16-s + 6·17-s + 18-s + 6·25-s − 8·26-s + 5·32-s + 6·34-s − 36-s + 8·43-s + 8·47-s + 6·49-s + 6·50-s + 8·52-s − 8·53-s − 8·59-s + 7·64-s − 6·68-s − 3·72-s + 81-s + 8·83-s + 8·86-s − 4·89-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s − 2.21·13-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 6/5·25-s − 1.56·26-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 1.21·43-s + 1.16·47-s + 6/7·49-s + 0.848·50-s + 1.10·52-s − 1.09·53-s − 1.04·59-s + 7/8·64-s − 0.727·68-s − 0.353·72-s + 1/9·81-s + 0.878·83-s + 0.862·86-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680485342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680485342\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.302483757388688250206219121702, −8.867859038834700445783886728292, −8.166653350987207092171350527951, −7.70039877615018488608822904718, −7.26772206528982663919449199617, −6.85939235744957576768978881780, −6.04962307317653729368284890078, −5.59683083656624915106383976380, −5.13376287434716239365793094807, −4.57131759146945849700598306856, −4.29021162045042281478729633150, −3.32025751641421918267197012068, −2.93877035683966351559302967234, −2.15002278014486058242440090009, −0.77745204865944098635796831973,
0.77745204865944098635796831973, 2.15002278014486058242440090009, 2.93877035683966351559302967234, 3.32025751641421918267197012068, 4.29021162045042281478729633150, 4.57131759146945849700598306856, 5.13376287434716239365793094807, 5.59683083656624915106383976380, 6.04962307317653729368284890078, 6.85939235744957576768978881780, 7.26772206528982663919449199617, 7.70039877615018488608822904718, 8.166653350987207092171350527951, 8.867859038834700445783886728292, 9.302483757388688250206219121702
