| L(s) = 1 | − 8·7-s − 6·9-s + 8·11-s − 4·13-s + 8·19-s + 4·23-s + 25-s − 4·29-s − 12·41-s − 16·43-s + 34·49-s + 48·63-s + 16·67-s − 12·73-s − 64·77-s + 27·81-s − 32·83-s + 32·91-s − 48·99-s + 12·101-s + 8·103-s + 24·117-s + 26·121-s + 127-s + 131-s − 64·133-s + 137-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2·9-s + 2.41·11-s − 1.10·13-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 0.742·29-s − 1.87·41-s − 2.43·43-s + 34/7·49-s + 6.04·63-s + 1.95·67-s − 1.40·73-s − 7.29·77-s + 3·81-s − 3.51·83-s + 3.35·91-s − 4.82·99-s + 1.19·101-s + 0.788·103-s + 2.21·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4594574226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4594574226\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.79829320548176219169262692501, −6.99475397882498342232415771994, −6.97238544391138654785050907556, −6.62558138018503710956136844851, −6.24669891559074187982136172346, −5.70093888866808737514223874840, −5.43937453001130315626300359266, −4.88100235382297631822765199855, −4.05743554139496542188801747995, −3.36942016554960782770484692407, −3.26180587408034592610487247010, −3.13138191848965214801586024221, −2.31591636420991224655303902281, −1.29463837983736440610705368799, −0.29435457435296185129512918262,
0.29435457435296185129512918262, 1.29463837983736440610705368799, 2.31591636420991224655303902281, 3.13138191848965214801586024221, 3.26180587408034592610487247010, 3.36942016554960782770484692407, 4.05743554139496542188801747995, 4.88100235382297631822765199855, 5.43937453001130315626300359266, 5.70093888866808737514223874840, 6.24669891559074187982136172346, 6.62558138018503710956136844851, 6.97238544391138654785050907556, 6.99475397882498342232415771994, 7.79829320548176219169262692501
