| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·9-s + 5·16-s + 8·18-s − 12·23-s + 12·29-s − 6·32-s − 12·36-s − 12·37-s − 24·43-s + 24·46-s + 12·53-s − 24·58-s + 7·64-s + 12·71-s + 16·72-s + 24·74-s − 20·79-s + 7·81-s + 48·86-s − 36·92-s − 24·106-s − 4·109-s − 36·113-s + 36·116-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 4/3·9-s + 5/4·16-s + 1.88·18-s − 2.50·23-s + 2.22·29-s − 1.06·32-s − 2·36-s − 1.97·37-s − 3.65·43-s + 3.53·46-s + 1.64·53-s − 3.15·58-s + 7/8·64-s + 1.42·71-s + 1.88·72-s + 2.78·74-s − 2.25·79-s + 7/9·81-s + 5.17·86-s − 3.75·92-s − 2.33·106-s − 0.383·109-s − 3.38·113-s + 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.660153886440728742280980086482, −8.337446602544081575207299536413, −8.138046882184356814328094691722, −7.995605997151710897062972608912, −7.14980456080731587847550912650, −6.95779844084293422933398018527, −6.37499198610427293486799143533, −6.37248260189936744351386468251, −5.66454141796563069222697288842, −5.37455954323228912946435291428, −4.94445039296623239893984358188, −4.29923574616628384446489828272, −3.53808758548069267701950775879, −3.47906817582929731367555338216, −2.57842518693119346484393496352, −2.47634928765299981144971801516, −1.71243919208373677478348813799, −1.23112608141791982397204180843, 0, 0,
1.23112608141791982397204180843, 1.71243919208373677478348813799, 2.47634928765299981144971801516, 2.57842518693119346484393496352, 3.47906817582929731367555338216, 3.53808758548069267701950775879, 4.29923574616628384446489828272, 4.94445039296623239893984358188, 5.37455954323228912946435291428, 5.66454141796563069222697288842, 6.37248260189936744351386468251, 6.37499198610427293486799143533, 6.95779844084293422933398018527, 7.14980456080731587847550912650, 7.995605997151710897062972608912, 8.138046882184356814328094691722, 8.337446602544081575207299536413, 8.660153886440728742280980086482
