L(s) = 1 | − 2-s + 4-s − 8-s − 9-s + 8·11-s + 16-s + 18-s − 8·22-s + 6·25-s − 32-s − 36-s − 8·43-s + 8·44-s − 7·49-s − 6·50-s + 64-s + 8·67-s + 72-s + 81-s + 8·86-s − 8·88-s + 7·98-s − 8·99-s + 6·100-s + 8·107-s + 28·113-s + 26·121-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 0.235·18-s − 1.70·22-s + 6/5·25-s − 0.176·32-s − 1/6·36-s − 1.21·43-s + 1.20·44-s − 49-s − 0.848·50-s + 1/8·64-s + 0.977·67-s + 0.117·72-s + 1/9·81-s + 0.862·86-s − 0.852·88-s + 0.707·98-s − 0.804·99-s + 3/5·100-s + 0.773·107-s + 2.63·113-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.136111527\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136111527\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.813536793719370156136934900670, −9.488830107252339252415216505294, −8.981242332530629627788661666863, −8.557041619171066391985106539338, −8.196264624071478486989468561724, −7.30425844399090075443129656281, −6.90505066176035016073039144973, −6.41053071455681571589224496085, −6.03012130094571440200692388332, −5.11719413976785294520251081011, −4.46475070828206177816117117812, −3.66600491223085146113555275917, −3.13759619746903620995726706166, −1.97494286469388269114343733065, −1.09691121476639347093183060072,
1.09691121476639347093183060072, 1.97494286469388269114343733065, 3.13759619746903620995726706166, 3.66600491223085146113555275917, 4.46475070828206177816117117812, 5.11719413976785294520251081011, 6.03012130094571440200692388332, 6.41053071455681571589224496085, 6.90505066176035016073039144973, 7.30425844399090075443129656281, 8.196264624071478486989468561724, 8.557041619171066391985106539338, 8.981242332530629627788661666863, 9.488830107252339252415216505294, 9.813536793719370156136934900670