| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·9-s + 4·11-s − 14-s + 16-s + 2·18-s − 4·22-s + 8·23-s − 25-s + 28-s − 32-s − 2·36-s − 8·37-s − 12·43-s + 4·44-s − 8·46-s + 49-s + 50-s − 56-s − 2·63-s + 64-s − 12·67-s + 2·72-s + 8·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 0.852·22-s + 1.66·23-s − 1/5·25-s + 0.188·28-s − 0.176·32-s − 1/3·36-s − 1.31·37-s − 1.82·43-s + 0.603·44-s − 1.17·46-s + 1/7·49-s + 0.141·50-s − 0.133·56-s − 0.251·63-s + 1/8·64-s − 1.46·67-s + 0.235·72-s + 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1097600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1097600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 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
−7.991402050078446769861497385292, −7.39062492414349481527228043076, −6.99794165751067222668429363386, −6.62008107217115123130684512433, −6.28796554336450030136081276811, −5.64158626495782784571154960578, −5.15674283001076257651240115180, −4.84718448015438342840988029616, −4.05373576190811202124463454879, −3.59465157011995269941145314091, −3.03133940350318595018343900156, −2.50256462084336627908097865240, −1.59252752795581237732790491504, −1.23802741421811599975391610130, 0,
1.23802741421811599975391610130, 1.59252752795581237732790491504, 2.50256462084336627908097865240, 3.03133940350318595018343900156, 3.59465157011995269941145314091, 4.05373576190811202124463454879, 4.84718448015438342840988029616, 5.15674283001076257651240115180, 5.64158626495782784571154960578, 6.28796554336450030136081276811, 6.62008107217115123130684512433, 6.99794165751067222668429363386, 7.39062492414349481527228043076, 7.991402050078446769861497385292
