| L(s) = 1 | − 4·3-s + 4-s + 6·9-s − 4·12-s − 4·13-s + 16-s + 12·17-s − 10·25-s + 4·27-s − 12·29-s + 6·36-s + 16·39-s + 16·43-s − 4·48-s + 49-s − 48·51-s − 4·52-s + 12·53-s + 16·61-s + 64-s + 12·68-s + 40·75-s + 16·79-s − 37·81-s + 48·87-s − 10·100-s − 8·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s − 1.10·13-s + 1/4·16-s + 2.91·17-s − 2·25-s + 0.769·27-s − 2.22·29-s + 36-s + 2.56·39-s + 2.43·43-s − 0.577·48-s + 1/7·49-s − 6.72·51-s − 0.554·52-s + 1.64·53-s + 2.04·61-s + 1/8·64-s + 1.45·68-s + 4.61·75-s + 1.80·79-s − 4.11·81-s + 5.14·87-s − 100-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5443988515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5443988515\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.49592006325651541786194815082, −10.05203493927597549730917438719, −9.765547119459919407856461234632, −9.048963436684934837350190246147, −7.985543387818251367130758863278, −7.57571100088867902110310233811, −7.21314366054207864456422661616, −6.39830303827760347609498636850, −5.67659105640301163348975864880, −5.57928681742950427486583645839, −5.32009626342602375518070026112, −4.20093704083038633652643494480, −3.42971528113344384838316462715, −2.23652265123208151083734264406, −0.78150816281722635549710851356,
0.78150816281722635549710851356, 2.23652265123208151083734264406, 3.42971528113344384838316462715, 4.20093704083038633652643494480, 5.32009626342602375518070026112, 5.57928681742950427486583645839, 5.67659105640301163348975864880, 6.39830303827760347609498636850, 7.21314366054207864456422661616, 7.57571100088867902110310233811, 7.985543387818251367130758863278, 9.048963436684934837350190246147, 9.765547119459919407856461234632, 10.05203493927597549730917438719, 10.49592006325651541786194815082
