| L(s) = 1 | + 2-s + 3-s + 6-s + 5·7-s − 8-s − 3·11-s − 5·13-s + 5·14-s − 16-s − 4·17-s + 19-s + 5·21-s − 3·22-s − 24-s − 5·26-s − 27-s − 2·29-s + 8·31-s − 3·33-s − 4·34-s − 9·37-s + 38-s − 5·39-s − 10·41-s + 5·42-s + 12·43-s + 14·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.88·7-s − 0.353·8-s − 0.904·11-s − 1.38·13-s + 1.33·14-s − 1/4·16-s − 0.970·17-s + 0.229·19-s + 1.09·21-s − 0.639·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.522·33-s − 0.685·34-s − 1.47·37-s + 0.162·38-s − 0.800·39-s − 1.56·41-s + 0.771·42-s + 1.82·43-s + 2.04·47-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
| \(L(1)\) |
\(\approx\) |
\(3.688866936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.688866936\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.317636194589072592332131285607, −8.813262476445423710104998593689, −8.530361958193510058691477743445, −8.270096655228454330662176349017, −7.82712271063298268213009461781, −7.48966930315583373404754209166, −7.01233634917083061982886758406, −6.86986756354231760645299476249, −5.95489223922732014938844080309, −5.70633171521605535772434893605, −5.09012390005828146631794790447, −4.93555865320630099128932942084, −4.63057964598419846905838938294, −4.12309185787924650042815987711, −3.66026798706937800600802147301, −2.96722172372363430859148666157, −2.34180151860474063013342118265, −2.30758922471590796290988658394, −1.57148287432617671596693334688, −0.57875239758375931385709083781,
0.57875239758375931385709083781, 1.57148287432617671596693334688, 2.30758922471590796290988658394, 2.34180151860474063013342118265, 2.96722172372363430859148666157, 3.66026798706937800600802147301, 4.12309185787924650042815987711, 4.63057964598419846905838938294, 4.93555865320630099128932942084, 5.09012390005828146631794790447, 5.70633171521605535772434893605, 5.95489223922732014938844080309, 6.86986756354231760645299476249, 7.01233634917083061982886758406, 7.48966930315583373404754209166, 7.82712271063298268213009461781, 8.270096655228454330662176349017, 8.530361958193510058691477743445, 8.813262476445423710104998593689, 9.317636194589072592332131285607
