L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s + 12-s − 2·15-s + 16-s − 18-s + 10·19-s − 2·20-s − 9·23-s − 24-s + 2·25-s + 27-s − 2·29-s + 2·30-s − 32-s + 36-s − 10·38-s + 2·40-s − 17·43-s − 2·45-s + 9·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 2.29·19-s − 0.447·20-s − 1.87·23-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.176·32-s + 1/6·36-s − 1.62·38-s + 0.316·40-s − 2.59·43-s − 0.298·45-s + 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−8.400686874959112351869823457307, −7.74805342770738313094174283808, −7.45760711693594334816037873734, −7.29721977882789549118075926550, −6.65915648248178976648627868698, −5.87926835670308333331197925204, −5.72452906770656251735941428478, −4.84573715379450105806219671503, −4.44455839389224847947022513088, −3.62989960249383154197740978609, −3.38707619834676104223178091963, −2.78515579611076668028648170212, −1.91489478279785520241156841104, −1.24045645405291554819899930921, 0,
1.24045645405291554819899930921, 1.91489478279785520241156841104, 2.78515579611076668028648170212, 3.38707619834676104223178091963, 3.62989960249383154197740978609, 4.44455839389224847947022513088, 4.84573715379450105806219671503, 5.72452906770656251735941428478, 5.87926835670308333331197925204, 6.65915648248178976648627868698, 7.29721977882789549118075926550, 7.45760711693594334816037873734, 7.74805342770738313094174283808, 8.400686874959112351869823457307