| L(s) = 1 | − 2·3-s + 3·9-s − 6·11-s + 6·17-s − 2·19-s − 10·27-s + 12·33-s − 6·41-s − 20·43-s − 14·49-s − 12·51-s + 4·57-s + 12·59-s + 14·67-s + 2·73-s + 20·81-s − 18·83-s + 18·89-s + 20·97-s − 18·99-s − 6·107-s − 18·113-s + 11·121-s + 12·123-s + 127-s + 40·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.80·11-s + 1.45·17-s − 0.458·19-s − 1.92·27-s + 2.08·33-s − 0.937·41-s − 3.04·43-s − 2·49-s − 1.68·51-s + 0.529·57-s + 1.56·59-s + 1.71·67-s + 0.234·73-s + 20/9·81-s − 1.97·83-s + 1.90·89-s + 2.03·97-s − 1.80·99-s − 0.580·107-s − 1.69·113-s + 121-s + 1.08·123-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−7.939724002771569385501129680407, −7.52618259086168918070326595856, −7.02913034123628046649906047335, −6.84095417691906478478812419427, −6.30728773965330542984906644477, −6.17562070056370089250699011503, −5.51439306110800516733455869480, −5.32709264408628589391872086183, −5.16239684480750661246223332754, −4.81256220073585316426354063260, −4.36503149139871097234030572234, −3.73354091258120423012028578223, −3.30101000828655216554427594440, −3.26941960389057798465229049737, −2.36364980743719547641105914999, −2.08268284399654895804492999815, −1.51981137106982608364168941711, −0.952661276307377357255815268635, 0, 0,
0.952661276307377357255815268635, 1.51981137106982608364168941711, 2.08268284399654895804492999815, 2.36364980743719547641105914999, 3.26941960389057798465229049737, 3.30101000828655216554427594440, 3.73354091258120423012028578223, 4.36503149139871097234030572234, 4.81256220073585316426354063260, 5.16239684480750661246223332754, 5.32709264408628589391872086183, 5.51439306110800516733455869480, 6.17562070056370089250699011503, 6.30728773965330542984906644477, 6.84095417691906478478812419427, 7.02913034123628046649906047335, 7.52618259086168918070326595856, 7.939724002771569385501129680407
