L(s) = 1 | + 2·2-s + 2·4-s − 6·9-s + 8·11-s − 4·16-s + 2·17-s − 12·18-s − 6·19-s + 16·22-s − 25-s − 8·32-s + 4·34-s − 12·36-s − 12·38-s + 24·41-s − 22·43-s + 16·44-s + 2·49-s − 2·50-s − 10·59-s − 8·64-s − 18·67-s + 4·68-s + 20·73-s − 12·76-s + 27·81-s + 48·82-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 2·9-s + 2.41·11-s − 16-s + 0.485·17-s − 2.82·18-s − 1.37·19-s + 3.41·22-s − 1/5·25-s − 1.41·32-s + 0.685·34-s − 2·36-s − 1.94·38-s + 3.74·41-s − 3.35·43-s + 2.41·44-s + 2/7·49-s − 0.282·50-s − 1.30·59-s − 64-s − 2.19·67-s + 0.485·68-s + 2.34·73-s − 1.37·76-s + 3·81-s + 5.30·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2050624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2050624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 179 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.50347147105603888482747921142, −6.69315229163834238910355551443, −6.45121168829548939914972425933, −6.18510863780788689281989494770, −5.98005464987425808278445960199, −5.26927125289354544444142895540, −5.00803114900472857755791914675, −4.35623596118551311981433575243, −3.91679401412707670092471176413, −3.65455224933898053965548391576, −3.09073806655570782422086105798, −2.57742433461389850447314956434, −2.02799860211410953372109627190, −1.17016266815841417647623069936, 0,
1.17016266815841417647623069936, 2.02799860211410953372109627190, 2.57742433461389850447314956434, 3.09073806655570782422086105798, 3.65455224933898053965548391576, 3.91679401412707670092471176413, 4.35623596118551311981433575243, 5.00803114900472857755791914675, 5.26927125289354544444142895540, 5.98005464987425808278445960199, 6.18510863780788689281989494770, 6.45121168829548939914972425933, 6.69315229163834238910355551443, 7.50347147105603888482747921142