| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·9-s − 8·13-s − 14-s + 16-s + 2·18-s − 10·25-s + 8·26-s + 28-s − 8·31-s − 32-s − 2·36-s + 16·43-s − 24·47-s + 49-s + 10·50-s − 8·52-s − 56-s + 16·61-s + 8·62-s − 2·63-s + 64-s − 8·67-s + 2·72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 2.21·13-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 2·25-s + 1.56·26-s + 0.188·28-s − 1.43·31-s − 0.176·32-s − 1/3·36-s + 2.43·43-s − 3.50·47-s + 1/7·49-s + 1.41·50-s − 1.10·52-s − 0.133·56-s + 2.04·61-s + 1.01·62-s − 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.235·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.765547119459919407856461234632, −9.649484617797021810103268663298, −8.934540413043951629895618965043, −8.408228795468761224431793416064, −7.68957389720506083723150978195, −7.57571100088867902110310233811, −6.96175265373852036821224883683, −6.17368924384109906924363362759, −5.57928681742950427486583645839, −5.06701824847582369879735343431, −4.31547623044891473234039816601, −3.42443184461883656887541100921, −2.49739457868022289115900838847, −1.90996831082337002056034743405, 0,
1.90996831082337002056034743405, 2.49739457868022289115900838847, 3.42443184461883656887541100921, 4.31547623044891473234039816601, 5.06701824847582369879735343431, 5.57928681742950427486583645839, 6.17368924384109906924363362759, 6.96175265373852036821224883683, 7.57571100088867902110310233811, 7.68957389720506083723150978195, 8.408228795468761224431793416064, 8.934540413043951629895618965043, 9.649484617797021810103268663298, 9.765547119459919407856461234632
