| L(s) = 1 | − 2·2-s + 2·4-s − 7-s − 4·9-s − 9·11-s + 2·14-s − 4·16-s + 8·18-s + 18·22-s + 4·23-s + 2·25-s − 2·28-s + 6·29-s + 8·32-s − 8·36-s − 12·37-s − 5·43-s − 18·44-s − 8·46-s − 6·49-s − 4·50-s − 15·53-s − 12·58-s + 4·63-s − 8·64-s − 9·67-s + 24·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s − 4/3·9-s − 2.71·11-s + 0.534·14-s − 16-s + 1.88·18-s + 3.83·22-s + 0.834·23-s + 2/5·25-s − 0.377·28-s + 1.11·29-s + 1.41·32-s − 4/3·36-s − 1.97·37-s − 0.762·43-s − 2.71·44-s − 1.17·46-s − 6/7·49-s − 0.565·50-s − 2.06·53-s − 1.57·58-s + 0.503·63-s − 64-s − 1.09·67-s + 2.78·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
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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
−7.67335078497563088571091378899, −7.34541246957182503630415572812, −6.83498527129197968027715848030, −6.31267361837488764541791580866, −5.92040950196015795642402819102, −5.18741352189353919318546131999, −4.97798032650342694038900051396, −4.71033769091101380316359591352, −3.51887916480050099952252418650, −3.06192101833268841070385132312, −2.69224722252436758931654889310, −2.18035164361485404411932544620, −1.28053725312198844601612391556, 0, 0,
1.28053725312198844601612391556, 2.18035164361485404411932544620, 2.69224722252436758931654889310, 3.06192101833268841070385132312, 3.51887916480050099952252418650, 4.71033769091101380316359591352, 4.97798032650342694038900051396, 5.18741352189353919318546131999, 5.92040950196015795642402819102, 6.31267361837488764541791580866, 6.83498527129197968027715848030, 7.34541246957182503630415572812, 7.67335078497563088571091378899
