L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s − 9-s + 4·10-s − 6·13-s + 16-s − 5·17-s + 18-s − 4·20-s + 11·25-s + 6·26-s + 2·29-s − 32-s + 5·34-s − 36-s − 7·37-s + 4·40-s − 4·41-s + 4·45-s + 10·49-s − 11·50-s − 6·52-s − 9·53-s − 2·58-s + 8·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s − 1/3·9-s + 1.26·10-s − 1.66·13-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.894·20-s + 11/5·25-s + 1.17·26-s + 0.371·29-s − 0.176·32-s + 0.857·34-s − 1/6·36-s − 1.15·37-s + 0.632·40-s − 0.624·41-s + 0.596·45-s + 10/7·49-s − 1.55·50-s − 0.832·52-s − 1.23·53-s − 0.262·58-s + 1.02·61-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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−7.74595910446197541695880798232, −7.47257520441391347710902337064, −6.96640055706376197643052591334, −6.86798482856206347581045986137, −6.21668245274537797842527557946, −5.56444121740678121118665607769, −4.95466299406284717989722375294, −4.62473460398657224821109691055, −4.21076801301995338693018618038, −3.48295794691699500896369178900, −3.12894602333861848418738184530, −2.44189137127608975975427283217, −1.91390592089216370115091186822, −0.67817962872893024981629975774, 0,
0.67817962872893024981629975774, 1.91390592089216370115091186822, 2.44189137127608975975427283217, 3.12894602333861848418738184530, 3.48295794691699500896369178900, 4.21076801301995338693018618038, 4.62473460398657224821109691055, 4.95466299406284717989722375294, 5.56444121740678121118665607769, 6.21668245274537797842527557946, 6.86798482856206347581045986137, 6.96640055706376197643052591334, 7.47257520441391347710902337064, 7.74595910446197541695880798232