L(s) = 1 | + 2-s + 4-s + 6·7-s + 8-s + 6·14-s + 16-s − 2·25-s + 6·28-s + 2·29-s + 32-s + 4·41-s + 12·43-s + 14·49-s − 2·50-s − 4·53-s + 6·56-s + 2·58-s + 18·59-s + 6·61-s + 64-s − 22·71-s − 8·73-s + 4·82-s + 12·86-s − 14·89-s + 14·98-s − 2·100-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.26·7-s + 0.353·8-s + 1.60·14-s + 1/4·16-s − 2/5·25-s + 1.13·28-s + 0.371·29-s + 0.176·32-s + 0.624·41-s + 1.82·43-s + 2·49-s − 0.282·50-s − 0.549·53-s + 0.801·56-s + 0.262·58-s + 2.34·59-s + 0.768·61-s + 1/8·64-s − 2.61·71-s − 0.936·73-s + 0.441·82-s + 1.29·86-s − 1.48·89-s + 1.41·98-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.247689036\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.247689036\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.429978719933034936568180255436, −8.118186073957973407998071681269, −7.54350751472713997928998850130, −7.32880707795846742344350884570, −6.77334843517255215491565017150, −6.01085360754429001735054476561, −5.69151698902320713689597419502, −5.18986672553277247243223026201, −4.75440863301217278744018278881, −4.19714967890496152051534596723, −4.01108706975301476570599849062, −2.98566801135777260476709204190, −2.40422723659045935235687479093, −1.75286816317943085884342132217, −1.10373413815389018714980648056,
1.10373413815389018714980648056, 1.75286816317943085884342132217, 2.40422723659045935235687479093, 2.98566801135777260476709204190, 4.01108706975301476570599849062, 4.19714967890496152051534596723, 4.75440863301217278744018278881, 5.18986672553277247243223026201, 5.69151698902320713689597419502, 6.01085360754429001735054476561, 6.77334843517255215491565017150, 7.32880707795846742344350884570, 7.54350751472713997928998850130, 8.118186073957973407998071681269, 8.429978719933034936568180255436