| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s − 4·12-s − 8·13-s − 4·16-s + 6·18-s − 16·26-s − 4·27-s + 4·31-s − 8·32-s + 6·36-s − 16·37-s + 16·39-s + 4·41-s − 8·43-s + 8·48-s + 10·49-s − 16·52-s + 12·53-s − 8·54-s + 8·62-s − 8·64-s + 24·67-s + 24·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s − 1.15·12-s − 2.21·13-s − 16-s + 1.41·18-s − 3.13·26-s − 0.769·27-s + 0.718·31-s − 1.41·32-s + 36-s − 2.63·37-s + 2.56·39-s + 0.624·41-s − 1.21·43-s + 1.15·48-s + 10/7·49-s − 2.21·52-s + 1.64·53-s − 1.08·54-s + 1.01·62-s − 64-s + 2.93·67-s + 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.690659418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690659418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−11.37331684323367922348054912130, −10.42438477100925057134788537796, −10.16937877639622175159315053319, −9.829932887489820091776894785556, −9.236559307019962860025346071624, −8.743380302330356769452213775578, −8.154389238982362704566127887242, −7.39455817417441250097264741188, −7.16239952312079613426910753772, −6.69363539304346412974409166717, −6.33381802077517724086867331514, −5.61505272819803059433681440237, −5.27213745154939131407998913986, −4.80713801216718580854087167474, −4.74817072293679982607834694681, −3.66080884828571460979894058315, −3.59109641256330707808232607961, −2.27687552175324204583534412412, −2.26658366836664787218179475693, −0.59793390921578088501730731337,
0.59793390921578088501730731337, 2.26658366836664787218179475693, 2.27687552175324204583534412412, 3.59109641256330707808232607961, 3.66080884828571460979894058315, 4.74817072293679982607834694681, 4.80713801216718580854087167474, 5.27213745154939131407998913986, 5.61505272819803059433681440237, 6.33381802077517724086867331514, 6.69363539304346412974409166717, 7.16239952312079613426910753772, 7.39455817417441250097264741188, 8.154389238982362704566127887242, 8.743380302330356769452213775578, 9.236559307019962860025346071624, 9.829932887489820091776894785556, 10.16937877639622175159315053319, 10.42438477100925057134788537796, 11.37331684323367922348054912130
