| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·9-s + 6·11-s − 2·14-s + 16-s + 2·18-s − 6·22-s − 12·23-s + 2·25-s + 2·28-s + 5·29-s − 32-s − 2·36-s + 4·37-s + 10·43-s + 6·44-s + 12·46-s − 3·49-s − 2·50-s − 2·56-s − 5·58-s − 4·63-s + 64-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 1.80·11-s − 0.534·14-s + 1/4·16-s + 0.471·18-s − 1.27·22-s − 2.50·23-s + 2/5·25-s + 0.377·28-s + 0.928·29-s − 0.176·32-s − 1/3·36-s + 0.657·37-s + 1.52·43-s + 0.904·44-s + 1.76·46-s − 3/7·49-s − 0.282·50-s − 0.267·56-s − 0.656·58-s − 0.503·63-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8237602295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8237602295\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−11.49162013882750967841183279965, −10.90158995958577256707144847412, −10.18796705127681919813296731841, −9.729819280511022086396866603247, −9.078023741838738412843467655147, −8.604890826682974059594363357622, −8.088501381964329434366366343272, −7.53849855219366047837847268040, −6.67446594677737454264973881731, −6.17781167218693040095529945209, −5.58634883710742549408139419861, −4.42259164830999812227433451708, −3.88784538082633044424758022151, −2.62855167833738310808009440358, −1.47603459749635641252603636631,
1.47603459749635641252603636631, 2.62855167833738310808009440358, 3.88784538082633044424758022151, 4.42259164830999812227433451708, 5.58634883710742549408139419861, 6.17781167218693040095529945209, 6.67446594677737454264973881731, 7.53849855219366047837847268040, 8.088501381964329434366366343272, 8.604890826682974059594363357622, 9.078023741838738412843467655147, 9.729819280511022086396866603247, 10.18796705127681919813296731841, 10.90158995958577256707144847412, 11.49162013882750967841183279965
