| L(s) = 1 | − 3·3-s + 2·5-s − 4·7-s + 6·9-s − 6·15-s − 6·17-s + 12·21-s + 3·25-s − 9·27-s − 8·35-s − 16·37-s − 12·41-s − 20·43-s + 12·45-s + 6·47-s + 9·49-s + 18·51-s + 12·59-s − 24·63-s − 4·67-s − 9·75-s + 26·79-s + 9·81-s − 24·83-s − 12·85-s − 36·101-s + 24·105-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 1.51·7-s + 2·9-s − 1.54·15-s − 1.45·17-s + 2.61·21-s + 3/5·25-s − 1.73·27-s − 1.35·35-s − 2.63·37-s − 1.87·41-s − 3.04·43-s + 1.78·45-s + 0.875·47-s + 9/7·49-s + 2.52·51-s + 1.56·59-s − 3.02·63-s − 0.488·67-s − 1.03·75-s + 2.92·79-s + 81-s − 2.63·83-s − 1.30·85-s − 3.58·101-s + 2.34·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 191 T^{2} + p^{2} T^{4} \) |
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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
−9.207335126577778234210436754797, −8.950341252529072309023142958660, −8.372212101040544387181389916528, −8.108819092954677713681880381354, −6.98663804602602396455899831436, −6.91949328359894455700780586332, −6.71102416352581825356665247875, −6.55630002074594847566235363337, −5.80172468740145864564357145507, −5.60437374517150491310183642870, −5.02307140227628708843292456111, −4.97230491107338122693678167620, −4.15566336632912677072328438588, −3.68157062070953272309657341271, −3.22328748958980064535225211317, −2.49598435586195513805787709841, −1.83926102115938740426825491224, −1.31473898481767415083221791176, 0, 0,
1.31473898481767415083221791176, 1.83926102115938740426825491224, 2.49598435586195513805787709841, 3.22328748958980064535225211317, 3.68157062070953272309657341271, 4.15566336632912677072328438588, 4.97230491107338122693678167620, 5.02307140227628708843292456111, 5.60437374517150491310183642870, 5.80172468740145864564357145507, 6.55630002074594847566235363337, 6.71102416352581825356665247875, 6.91949328359894455700780586332, 6.98663804602602396455899831436, 8.108819092954677713681880381354, 8.372212101040544387181389916528, 8.950341252529072309023142958660, 9.207335126577778234210436754797
