L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 6·9-s − 4·14-s − 16-s + 6·18-s − 8·19-s − 6·25-s − 4·28-s + 12·29-s + 8·31-s − 5·32-s + 6·36-s − 4·37-s + 8·38-s + 9·49-s + 6·50-s + 12·53-s + 12·56-s − 12·58-s − 24·59-s − 8·62-s − 24·63-s + 7·64-s − 18·72-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 2·9-s − 1.06·14-s − 1/4·16-s + 1.41·18-s − 1.83·19-s − 6/5·25-s − 0.755·28-s + 2.22·29-s + 1.43·31-s − 0.883·32-s + 36-s − 0.657·37-s + 1.29·38-s + 9/7·49-s + 0.848·50-s + 1.64·53-s + 1.60·56-s − 1.57·58-s − 3.12·59-s − 1.01·62-s − 3.02·63-s + 7/8·64-s − 2.12·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.695686711870280818326611584019, −8.454477830081832051786301929374, −7.925296238092951843585506881800, −7.81910395523808377988054564239, −6.93074374025885353931015118815, −6.20029722030088610404284024481, −5.99045132204207329080028903613, −5.20530356371725330671819659135, −4.74199315541377008016560012773, −4.42300131830897794189074263961, −3.66457291810612686968408552991, −2.70885064487133489617707672764, −2.20828403093183834517943119600, −1.23410526437737987553968393792, 0,
1.23410526437737987553968393792, 2.20828403093183834517943119600, 2.70885064487133489617707672764, 3.66457291810612686968408552991, 4.42300131830897794189074263961, 4.74199315541377008016560012773, 5.20530356371725330671819659135, 5.99045132204207329080028903613, 6.20029722030088610404284024481, 6.93074374025885353931015118815, 7.81910395523808377988054564239, 7.925296238092951843585506881800, 8.454477830081832051786301929374, 8.695686711870280818326611584019