| L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s + 2·11-s + 16-s − 2·18-s − 2·22-s − 4·23-s + 2·25-s + 2·29-s − 32-s + 2·36-s − 4·37-s + 8·43-s + 2·44-s + 4·46-s − 7·49-s − 2·50-s − 10·53-s − 2·58-s + 64-s − 6·67-s + 4·71-s − 2·72-s + 4·74-s − 5·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 0.603·11-s + 1/4·16-s − 0.471·18-s − 0.426·22-s − 0.834·23-s + 2/5·25-s + 0.371·29-s − 0.176·32-s + 1/3·36-s − 0.657·37-s + 1.21·43-s + 0.301·44-s + 0.589·46-s − 49-s − 0.282·50-s − 1.37·53-s − 0.262·58-s + 1/8·64-s − 0.733·67-s + 0.474·71-s − 0.235·72-s + 0.464·74-s − 5/9·81-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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 41 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75879282270896903377551228368, −7.39578678415885792104275519763, −6.97556319921434535593897623868, −6.40638575684129168429288779053, −6.27064936742835988224087556021, −5.68452106093720067685647453040, −5.00240665962561188704553339995, −4.69328291528169407241091839361, −3.93221214628784589867356016281, −3.73297582042354673548602632973, −2.90985414229579052886258136156, −2.42217906983837693240322077516, −1.59618459644696579161799921540, −1.20477784251621165121279381575, 0,
1.20477784251621165121279381575, 1.59618459644696579161799921540, 2.42217906983837693240322077516, 2.90985414229579052886258136156, 3.73297582042354673548602632973, 3.93221214628784589867356016281, 4.69328291528169407241091839361, 5.00240665962561188704553339995, 5.68452106093720067685647453040, 6.27064936742835988224087556021, 6.40638575684129168429288779053, 6.97556319921434535593897623868, 7.39578678415885792104275519763, 7.75879282270896903377551228368
