| L(s) = 1 | − 4-s − 2·7-s − 2·9-s − 8·13-s − 3·16-s − 6·23-s + 25-s + 2·28-s + 6·29-s + 2·36-s − 2·49-s + 8·52-s + 12·59-s + 4·63-s + 7·64-s + 10·67-s − 12·71-s − 5·81-s − 18·83-s + 16·91-s + 6·92-s − 100-s + 10·103-s − 30·107-s + 16·109-s + 6·112-s − 6·116-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s − 2/3·9-s − 2.21·13-s − 3/4·16-s − 1.25·23-s + 1/5·25-s + 0.377·28-s + 1.11·29-s + 1/3·36-s − 2/7·49-s + 1.10·52-s + 1.56·59-s + 0.503·63-s + 7/8·64-s + 1.22·67-s − 1.42·71-s − 5/9·81-s − 1.97·83-s + 1.67·91-s + 0.625·92-s − 0.0999·100-s + 0.985·103-s − 2.90·107-s + 1.53·109-s + 0.566·112-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.31021475504937948132104477448, −9.885987153651444884556648357228, −9.696396268952817799509713689061, −8.942926438758517872552270660149, −8.471425133280778904402131824110, −7.85912181863133156548032228725, −7.12722301011653113774252073602, −6.72933335047920367945929344948, −5.95537804679188128605938024551, −5.26242766216616259639501937046, −4.66787191806607434412184181990, −4.00198380042866898411922941051, −2.91417830661177268148435641391, −2.32769573124223567048228946598, 0,
2.32769573124223567048228946598, 2.91417830661177268148435641391, 4.00198380042866898411922941051, 4.66787191806607434412184181990, 5.26242766216616259639501937046, 5.95537804679188128605938024551, 6.72933335047920367945929344948, 7.12722301011653113774252073602, 7.85912181863133156548032228725, 8.471425133280778904402131824110, 8.942926438758517872552270660149, 9.696396268952817799509713689061, 9.885987153651444884556648357228, 10.31021475504937948132104477448
