| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 8·11-s − 7·16-s + 16·22-s + 25-s + 4·29-s + 14·32-s − 20·37-s + 8·43-s − 8·44-s − 7·49-s + 2·50-s + 20·53-s + 8·58-s + 35·64-s + 24·67-s + 16·71-s − 40·74-s + 16·86-s − 64·88-s − 14·98-s − 100-s + 40·106-s + 24·107-s + 28·109-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 2.41·11-s − 7/4·16-s + 3.41·22-s + 1/5·25-s + 0.742·29-s + 2.47·32-s − 3.28·37-s + 1.21·43-s − 1.20·44-s − 49-s + 0.282·50-s + 2.74·53-s + 1.05·58-s + 35/8·64-s + 2.93·67-s + 1.89·71-s − 4.64·74-s + 1.72·86-s − 6.82·88-s − 1.41·98-s − 0.0999·100-s + 3.88·106-s + 2.32·107-s + 2.68·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253375518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253375518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.730283045024072287400674871237, −8.890589428908360265391160029852, −8.627000477337622059216224790867, −8.554588868470699894187777077269, −7.37485778338999061621083536758, −6.76955885158912340897756950945, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.04860090093668311608273372623, −5.03119182641673168471146566272, −4.09922216243454775532088496022, −3.63412818236731287070311550143, −3.56571005686312005410318267112, −2.32940333880865099555943992805, −0.964820104627525128067217267144,
0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.56571005686312005410318267112, 3.63412818236731287070311550143, 4.09922216243454775532088496022, 5.03119182641673168471146566272, 5.04860090093668311608273372623, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 6.76955885158912340897756950945, 7.37485778338999061621083536758, 8.554588868470699894187777077269, 8.627000477337622059216224790867, 8.890589428908360265391160029852, 9.730283045024072287400674871237
