| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5·5-s + 2·6-s + 3·8-s + 3·9-s − 5·10-s + 2·12-s − 10·15-s − 16-s − 3·17-s − 3·18-s + 19-s − 5·20-s − 6·24-s + 11·25-s − 4·27-s + 10·30-s − 7·31-s − 5·32-s + 3·34-s − 3·36-s − 38-s + 15·40-s + 15·45-s + 2·48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 2.23·5-s + 0.816·6-s + 1.06·8-s + 9-s − 1.58·10-s + 0.577·12-s − 2.58·15-s − 1/4·16-s − 0.727·17-s − 0.707·18-s + 0.229·19-s − 1.11·20-s − 1.22·24-s + 11/5·25-s − 0.769·27-s + 1.82·30-s − 1.25·31-s − 0.883·32-s + 0.514·34-s − 1/2·36-s − 0.162·38-s + 2.37·40-s + 2.23·45-s + 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 165 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 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
−7.81947458032215673314917532689, −7.49134181637822409019182211683, −6.90820799466782018221444454712, −6.51538629427949879427875269097, −6.08900451382462637094878921273, −5.66001524938879741637465838582, −5.36968156418594919959726385528, −4.88323745463589915070432771723, −4.46103397436716246743238818421, −3.82221976196370896553630440730, −2.99753924749007578976059399541, −2.09439797072401209883470616174, −1.76289656293079236896734150247, −1.12946967656516647051616965624, 0,
1.12946967656516647051616965624, 1.76289656293079236896734150247, 2.09439797072401209883470616174, 2.99753924749007578976059399541, 3.82221976196370896553630440730, 4.46103397436716246743238818421, 4.88323745463589915070432771723, 5.36968156418594919959726385528, 5.66001524938879741637465838582, 6.08900451382462637094878921273, 6.51538629427949879427875269097, 6.90820799466782018221444454712, 7.49134181637822409019182211683, 7.81947458032215673314917532689
