| L(s) = 1 | − 2-s + 4-s + 5-s + 8-s + 9-s − 10-s − 3·11-s − 8·13-s − 3·16-s − 18-s − 4·19-s + 20-s + 3·22-s + 2·23-s − 3·25-s + 8·26-s − 2·29-s + 5·32-s + 36-s + 4·38-s + 40-s − 10·41-s − 3·44-s + 45-s − 2·46-s + 4·47-s − 11·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 2.21·13-s − 3/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.639·22-s + 0.417·23-s − 3/5·25-s + 1.56·26-s − 0.371·29-s + 0.883·32-s + 1/6·36-s + 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.452·44-s + 0.149·45-s − 0.294·46-s + 0.583·47-s − 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
| 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
−9.450223779291344770807595463754, −9.140781653340821274478576792169, −8.246617768836988210813697607109, −7.996925043161845275175381485328, −7.47472357059831168214977383824, −6.91320625381387899247441931584, −6.66737315612773205927859990214, −5.77349495078356174773613983416, −5.09725966565904174656575072428, −4.79862490565530800982393900729, −4.07549449166981345454199458144, −2.97385313027726012133994344346, −2.31538745683713410959810010695, −1.75487743553433853867216320915, 0,
1.75487743553433853867216320915, 2.31538745683713410959810010695, 2.97385313027726012133994344346, 4.07549449166981345454199458144, 4.79862490565530800982393900729, 5.09725966565904174656575072428, 5.77349495078356174773613983416, 6.66737315612773205927859990214, 6.91320625381387899247441931584, 7.47472357059831168214977383824, 7.996925043161845275175381485328, 8.246617768836988210813697607109, 9.140781653340821274478576792169, 9.450223779291344770807595463754
