| L(s) = 1 | + 2-s + 4-s − 5·7-s + 8-s − 5·14-s + 16-s − 9·17-s + 3·23-s + 2·25-s − 5·28-s − 11·31-s + 32-s − 9·34-s − 6·41-s + 3·46-s − 3·47-s + 7·49-s + 2·50-s − 5·56-s − 11·62-s + 64-s − 9·68-s − 27·71-s + 10·73-s + 7·79-s − 6·82-s + 3·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.88·7-s + 0.353·8-s − 1.33·14-s + 1/4·16-s − 2.18·17-s + 0.625·23-s + 2/5·25-s − 0.944·28-s − 1.97·31-s + 0.176·32-s − 1.54·34-s − 0.937·41-s + 0.442·46-s − 0.437·47-s + 49-s + 0.282·50-s − 0.668·56-s − 1.39·62-s + 1/8·64-s − 1.09·68-s − 3.20·71-s + 1.17·73-s + 0.787·79-s − 0.662·82-s + 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−9.318064584291740508655761445266, −8.961629232991490976348863058204, −8.603669782168897368906294678692, −7.67393294606200079253290035446, −7.09807394349229086907096533747, −6.76394884560908878105177246857, −6.33782224685832006548899547139, −5.88910934347139277423163203950, −5.10108455737906931058966039782, −4.59577871349919181010665541094, −3.82160231399275649953817271679, −3.34387034138904092881040240151, −2.71658131942521411027853473344, −1.88152557019066596749863474184, 0,
1.88152557019066596749863474184, 2.71658131942521411027853473344, 3.34387034138904092881040240151, 3.82160231399275649953817271679, 4.59577871349919181010665541094, 5.10108455737906931058966039782, 5.88910934347139277423163203950, 6.33782224685832006548899547139, 6.76394884560908878105177246857, 7.09807394349229086907096533747, 7.67393294606200079253290035446, 8.603669782168897368906294678692, 8.961629232991490976348863058204, 9.318064584291740508655761445266
