L(s) = 1 | − 4·3-s − 4·4-s − 6·5-s + 6·9-s + 16·12-s + 24·15-s + 12·16-s + 24·20-s + 6·23-s + 17·25-s + 4·27-s + 10·31-s − 24·36-s + 4·37-s − 36·45-s + 6·47-s − 48·48-s + 49-s − 18·53-s − 96·60-s − 32·64-s + 28·67-s − 24·69-s − 12·71-s − 68·75-s − 72·80-s − 37·81-s + ⋯ |
L(s) = 1 | − 2.30·3-s − 2·4-s − 2.68·5-s + 2·9-s + 4.61·12-s + 6.19·15-s + 3·16-s + 5.36·20-s + 1.25·23-s + 17/5·25-s + 0.769·27-s + 1.79·31-s − 4·36-s + 0.657·37-s − 5.36·45-s + 0.875·47-s − 6.92·48-s + 1/7·49-s − 2.47·53-s − 12.3·60-s − 4·64-s + 3.42·67-s − 2.88·69-s − 1.42·71-s − 7.85·75-s − 8.04·80-s − 4.11·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.023034081311283559976527256533, −7.60332298682950357021318080507, −6.97556725662558611504386870793, −6.48432181419363077598714474936, −6.05510665689561164799845565623, −5.47928828118502604201716906159, −4.95454046107923865998450494116, −4.82094210358198316082444203410, −4.36526056113723032255916840174, −4.04609643315305459582542236808, −3.36854503826664578071054317862, −3.00675363687601775920051816088, −0.883826827840201941525555392849, −0.71239963936312711175952583959, 0,
0.71239963936312711175952583959, 0.883826827840201941525555392849, 3.00675363687601775920051816088, 3.36854503826664578071054317862, 4.04609643315305459582542236808, 4.36526056113723032255916840174, 4.82094210358198316082444203410, 4.95454046107923865998450494116, 5.47928828118502604201716906159, 6.05510665689561164799845565623, 6.48432181419363077598714474936, 6.97556725662558611504386870793, 7.60332298682950357021318080507, 8.023034081311283559976527256533