| L(s) = 1 | − 2-s + 4-s − 8-s − 4·9-s − 2·11-s + 16-s + 4·18-s + 2·22-s − 2·23-s + 6·25-s + 2·29-s − 32-s − 4·36-s + 10·37-s − 18·43-s − 2·44-s + 2·46-s − 6·50-s + 8·53-s − 2·58-s + 64-s + 12·67-s − 12·71-s + 4·72-s − 10·74-s − 8·79-s + 7·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 4/3·9-s − 0.603·11-s + 1/4·16-s + 0.942·18-s + 0.426·22-s − 0.417·23-s + 6/5·25-s + 0.371·29-s − 0.176·32-s − 2/3·36-s + 1.64·37-s − 2.74·43-s − 0.301·44-s + 0.294·46-s − 0.848·50-s + 1.09·53-s − 0.262·58-s + 1/8·64-s + 1.46·67-s − 1.42·71-s + 0.471·72-s − 1.16·74-s − 0.900·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 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
−8.167952183117278656200361381964, −7.985214355894243268985951005897, −7.27482422787348251735601638905, −6.91476352071354127082003194560, −6.37006424054677819669056523615, −5.94886353815141496489163010243, −5.49159877330805004724472599985, −4.97094241930675111570599949060, −4.49990284235012122714804562060, −3.64914405573799547579917670177, −3.07784575310318789970718241706, −2.66361280218701772945112026393, −2.04391620963403611815649931466, −1.03137417045148774883228084324, 0,
1.03137417045148774883228084324, 2.04391620963403611815649931466, 2.66361280218701772945112026393, 3.07784575310318789970718241706, 3.64914405573799547579917670177, 4.49990284235012122714804562060, 4.97094241930675111570599949060, 5.49159877330805004724472599985, 5.94886353815141496489163010243, 6.37006424054677819669056523615, 6.91476352071354127082003194560, 7.27482422787348251735601638905, 7.985214355894243268985951005897, 8.167952183117278656200361381964
