| L(s) = 1 | + 2·2-s + 2·4-s + 8·7-s + 16·14-s − 4·16-s + 12·17-s + 16·23-s + 16·28-s − 4·31-s − 8·32-s + 24·34-s − 8·41-s + 32·46-s − 8·47-s + 34·49-s − 8·62-s − 8·64-s + 24·68-s − 16·71-s − 20·73-s − 4·79-s − 16·82-s + 8·89-s + 32·92-s − 16·94-s − 4·97-s + 68·98-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 3.02·7-s + 4.27·14-s − 16-s + 2.91·17-s + 3.33·23-s + 3.02·28-s − 0.718·31-s − 1.41·32-s + 4.11·34-s − 1.24·41-s + 4.71·46-s − 1.16·47-s + 34/7·49-s − 1.01·62-s − 64-s + 2.91·68-s − 1.89·71-s − 2.34·73-s − 0.450·79-s − 1.76·82-s + 0.847·89-s + 3.33·92-s − 1.65·94-s − 0.406·97-s + 6.86·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.727183947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.727183947\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.392721504234096432359990531208, −8.903426359490163661291248050705, −8.482028905972932003950734746220, −8.457096642259360064016038655407, −7.59138394013942229693116784555, −7.55385578392443887920688648977, −7.23724223899227357523869345801, −6.71223234170552535144499008753, −5.88738760623468955520421076903, −5.66937401854381764778651097601, −5.24927021820265879260334983042, −4.85887063653159332429535836449, −4.81661848869019682358419638026, −4.33579943522797220002256795134, −3.47546020473589403217086917997, −3.28287707232107419112779355760, −2.80444274782683784734615144393, −1.94857505010807483914855605461, −1.37572954293022635616934884168, −1.11651627375026001607750392742,
1.11651627375026001607750392742, 1.37572954293022635616934884168, 1.94857505010807483914855605461, 2.80444274782683784734615144393, 3.28287707232107419112779355760, 3.47546020473589403217086917997, 4.33579943522797220002256795134, 4.81661848869019682358419638026, 4.85887063653159332429535836449, 5.24927021820265879260334983042, 5.66937401854381764778651097601, 5.88738760623468955520421076903, 6.71223234170552535144499008753, 7.23724223899227357523869345801, 7.55385578392443887920688648977, 7.59138394013942229693116784555, 8.457096642259360064016038655407, 8.482028905972932003950734746220, 8.903426359490163661291248050705, 9.392721504234096432359990531208
