L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·14-s + 16-s − 3·23-s + 8·25-s − 2·28-s + 10·31-s − 32-s + 6·41-s + 3·46-s − 6·47-s − 2·49-s − 8·50-s + 2·56-s − 10·62-s + 64-s − 18·71-s + 4·73-s + 7·79-s − 6·82-s + 9·89-s − 3·92-s + 6·94-s − 11·97-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 0.625·23-s + 8/5·25-s − 0.377·28-s + 1.79·31-s − 0.176·32-s + 0.937·41-s + 0.442·46-s − 0.875·47-s − 2/7·49-s − 1.13·50-s + 0.267·56-s − 1.27·62-s + 1/8·64-s − 2.13·71-s + 0.468·73-s + 0.787·79-s − 0.662·82-s + 0.953·89-s − 0.312·92-s + 0.618·94-s − 1.11·97-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
\(L(1)\) |
\(\approx\) |
\(0.9389865857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9389865857\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.603384002109920925484166971787, −9.183673376102949056621935040645, −8.610028361000783704310897032066, −8.279133309389622600251427438967, −7.65274014464953476243705691750, −7.15783105795791171099208071963, −6.55789704255062500582585448580, −6.24150487485003178520648701843, −5.66650334980828507799304208986, −4.78611486436286309893354863873, −4.36075045479923626167253518031, −3.30513816527997452318395215051, −2.94741495006642520553938109985, −2.02243932175308410887290963272, −0.818618891798656203725184699877,
0.818618891798656203725184699877, 2.02243932175308410887290963272, 2.94741495006642520553938109985, 3.30513816527997452318395215051, 4.36075045479923626167253518031, 4.78611486436286309893354863873, 5.66650334980828507799304208986, 6.24150487485003178520648701843, 6.55789704255062500582585448580, 7.15783105795791171099208071963, 7.65274014464953476243705691750, 8.279133309389622600251427438967, 8.610028361000783704310897032066, 9.183673376102949056621935040645, 9.603384002109920925484166971787