| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 9-s + 2·10-s − 2·12-s + 15-s − 4·16-s − 2·18-s + 8·19-s − 2·20-s + 4·23-s + 25-s − 27-s − 6·29-s − 2·30-s + 8·32-s + 2·36-s − 16·38-s − 45-s − 8·46-s − 20·47-s + 4·48-s − 2·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s + 0.632·10-s − 0.577·12-s + 0.258·15-s − 16-s − 0.471·18-s + 1.83·19-s − 0.447·20-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.365·30-s + 1.41·32-s + 1/3·36-s − 2.59·38-s − 0.149·45-s − 1.17·46-s − 2.91·47-s + 0.577·48-s − 2/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−8.906881996248558700166656303194, −8.230572573815583958948491444344, −7.957466895878972708821372600445, −7.45803704054648868026646935427, −7.04449698469151449038363144300, −6.67360595941637015736441060871, −5.96875459139899375120191139193, −5.35920973493245625045695390529, −4.85995333200255810434699163290, −4.35487194595513828168582266802, −3.40979687775771901956269990315, −2.99015924018850445255115468534, −1.79048358137880339498803967465, −1.16882828116061610125844061576, 0,
1.16882828116061610125844061576, 1.79048358137880339498803967465, 2.99015924018850445255115468534, 3.40979687775771901956269990315, 4.35487194595513828168582266802, 4.85995333200255810434699163290, 5.35920973493245625045695390529, 5.96875459139899375120191139193, 6.67360595941637015736441060871, 7.04449698469151449038363144300, 7.45803704054648868026646935427, 7.957466895878972708821372600445, 8.230572573815583958948491444344, 8.906881996248558700166656303194
