L(s) = 1 | − 2·3-s + 4-s − 4·5-s + 3·9-s − 4·11-s − 2·12-s + 8·15-s + 16-s − 4·20-s + 16·23-s + 2·25-s − 4·27-s + 8·33-s + 3·36-s − 20·37-s − 4·44-s − 12·45-s − 2·48-s + 49-s + 12·53-s + 16·55-s + 8·59-s + 8·60-s + 64-s + 8·67-s − 32·69-s + 16·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.78·5-s + 9-s − 1.20·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.894·20-s + 3.33·23-s + 2/5·25-s − 0.769·27-s + 1.39·33-s + 1/2·36-s − 3.28·37-s − 0.603·44-s − 1.78·45-s − 0.288·48-s + 1/7·49-s + 1.64·53-s + 2.15·55-s + 1.04·59-s + 1.03·60-s + 1/8·64-s + 0.977·67-s − 3.85·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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
−8.552050058042110402630004221159, −8.356918966632091922574869118017, −7.73911516911369178381582752125, −7.20173970953094596472337873942, −6.84552480602986516155667793601, −6.83628576568467924985122855184, −5.59471126273469040252337343985, −5.33985014787602837985094112170, −4.99671622655233348150177845954, −4.26019449778485466667999987817, −3.62482887081886485478101246558, −3.18291183630288302233869260139, −2.28291322652895947414583514286, −1.03959629946639927298706906942, 0,
1.03959629946639927298706906942, 2.28291322652895947414583514286, 3.18291183630288302233869260139, 3.62482887081886485478101246558, 4.26019449778485466667999987817, 4.99671622655233348150177845954, 5.33985014787602837985094112170, 5.59471126273469040252337343985, 6.83628576568467924985122855184, 6.84552480602986516155667793601, 7.20173970953094596472337873942, 7.73911516911369178381582752125, 8.356918966632091922574869118017, 8.552050058042110402630004221159