| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·6-s + 8·7-s − 4·8-s + 9-s − 3·12-s − 16·14-s + 5·16-s − 2·18-s + 19-s − 8·21-s + 4·24-s − 10·25-s − 27-s + 24·28-s − 12·29-s − 6·32-s + 3·36-s − 2·38-s + 20·41-s + 16·42-s − 24·43-s − 5·48-s + 34·49-s + 20·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.816·6-s + 3.02·7-s − 1.41·8-s + 1/3·9-s − 0.866·12-s − 4.27·14-s + 5/4·16-s − 0.471·18-s + 0.229·19-s − 1.74·21-s + 0.816·24-s − 2·25-s − 0.192·27-s + 4.53·28-s − 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.324·38-s + 3.12·41-s + 2.46·42-s − 3.65·43-s − 0.721·48-s + 34/7·49-s + 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.202695186334477258022172533134, −7.64198238847393687882301889209, −7.47467831026680190924032557177, −7.09437019996928771238631970369, −6.23458925972747170402160537885, −5.74836164136915456195765077676, −5.47887733490393183920061047135, −4.96168584606520975721850935075, −4.27018118796272972112253469343, −4.03302788649080589834097790219, −2.97436876920201446806047330172, −2.06463630287918479320818570303, −1.65254561433729232279648150469, −1.37257859192148604205877773155, 0,
1.37257859192148604205877773155, 1.65254561433729232279648150469, 2.06463630287918479320818570303, 2.97436876920201446806047330172, 4.03302788649080589834097790219, 4.27018118796272972112253469343, 4.96168584606520975721850935075, 5.47887733490393183920061047135, 5.74836164136915456195765077676, 6.23458925972747170402160537885, 7.09437019996928771238631970369, 7.47467831026680190924032557177, 7.64198238847393687882301889209, 8.202695186334477258022172533134
