| L(s) = 1 | − 2·3-s + 4-s − 4·5-s + 3·9-s − 6·11-s − 2·12-s + 8·15-s + 16-s − 4·20-s − 2·23-s + 2·25-s − 4·27-s + 16·31-s + 12·33-s + 3·36-s − 6·44-s − 12·45-s − 16·47-s − 2·48-s − 10·49-s + 4·53-s + 24·55-s − 24·59-s + 8·60-s + 64-s − 24·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.78·5-s + 9-s − 1.80·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.894·20-s − 0.417·23-s + 2/5·25-s − 0.769·27-s + 2.87·31-s + 2.08·33-s + 1/2·36-s − 0.904·44-s − 1.78·45-s − 2.33·47-s − 0.288·48-s − 1.42·49-s + 0.549·53-s + 3.23·55-s − 3.12·59-s + 1.03·60-s + 1/8·64-s − 2.93·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.48954786032446015166944686975, −6.83453031171442160673681690396, −6.25250569926844252670184221360, −6.18048034541394024619090771603, −5.62488304743255639551871600674, −4.81553639281966938826542345391, −4.78570582427944795062347753374, −4.41468306928216671378208171036, −3.72941845560947568078388865858, −3.05418356833250210459565204108, −2.91631229938310047782795220333, −1.96782756403238297527837052883, −1.20358002063523422043845978105, 0, 0,
1.20358002063523422043845978105, 1.96782756403238297527837052883, 2.91631229938310047782795220333, 3.05418356833250210459565204108, 3.72941845560947568078388865858, 4.41468306928216671378208171036, 4.78570582427944795062347753374, 4.81553639281966938826542345391, 5.62488304743255639551871600674, 6.18048034541394024619090771603, 6.25250569926844252670184221360, 6.83453031171442160673681690396, 7.48954786032446015166944686975
