L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 5·17-s − 18-s + 25-s + 4·29-s − 32-s + 5·34-s + 36-s − 37-s − 16·41-s − 8·49-s − 50-s + 7·53-s − 4·58-s + 14·61-s + 64-s − 5·68-s − 72-s + 5·73-s + 74-s + 81-s + 16·82-s − 2·89-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1/5·25-s + 0.742·29-s − 0.176·32-s + 0.857·34-s + 1/6·36-s − 0.164·37-s − 2.49·41-s − 8/7·49-s − 0.141·50-s + 0.961·53-s − 0.525·58-s + 1.79·61-s + 1/8·64-s − 0.606·68-s − 0.117·72-s + 0.585·73-s + 0.116·74-s + 1/9·81-s + 1.76·82-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.81569857650125098439065885692, −7.41967893188920318352026767985, −6.82913000061957012372257152727, −6.59188427270805086448045504340, −6.38823544644026819052666029640, −5.49896144712689598127682265603, −5.20485142635874799828885762524, −4.66056503941909570449604283807, −4.13169647307162551588838959031, −3.55283617539668639272416985298, −3.02073178678657133571294320723, −2.30774136001233637346958815441, −1.85353470478522489447013864991, −1.04629605945313951688018210412, 0,
1.04629605945313951688018210412, 1.85353470478522489447013864991, 2.30774136001233637346958815441, 3.02073178678657133571294320723, 3.55283617539668639272416985298, 4.13169647307162551588838959031, 4.66056503941909570449604283807, 5.20485142635874799828885762524, 5.49896144712689598127682265603, 6.38823544644026819052666029640, 6.59188427270805086448045504340, 6.82913000061957012372257152727, 7.41967893188920318352026767985, 7.81569857650125098439065885692