| L(s) = 1 | − 3·3-s + 4-s − 6·7-s + 6·9-s − 3·12-s − 10·13-s + 16-s + 18·21-s − 25-s − 9·27-s − 6·28-s − 20·31-s + 6·36-s − 20·37-s + 30·39-s + 8·43-s − 3·48-s + 13·49-s − 10·52-s + 24·61-s − 36·63-s + 64-s − 16·67-s − 12·73-s + 3·75-s + 2·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s − 2.26·7-s + 2·9-s − 0.866·12-s − 2.77·13-s + 1/4·16-s + 3.92·21-s − 1/5·25-s − 1.73·27-s − 1.13·28-s − 3.59·31-s + 36-s − 3.28·37-s + 4.80·39-s + 1.21·43-s − 0.433·48-s + 13/7·49-s − 1.38·52-s + 3.07·61-s − 4.53·63-s + 1/8·64-s − 1.95·67-s − 1.40·73-s + 0.346·75-s + 0.225·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
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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
−8.820853343801771743435698779760, −7.66138306465432851890178277032, −7.17450713167885343099030707177, −6.93717508441912222809183241428, −6.92116149525582118764831835825, −5.95900466730196271449972715732, −5.69924671275910195432130153357, −5.22641373981494590954939993299, −4.76932431908835403735443234199, −3.80933035482911494201207656757, −3.43543530646035424633948117891, −2.56149928245518451776775111666, −1.83580907328715839320391152738, 0, 0,
1.83580907328715839320391152738, 2.56149928245518451776775111666, 3.43543530646035424633948117891, 3.80933035482911494201207656757, 4.76932431908835403735443234199, 5.22641373981494590954939993299, 5.69924671275910195432130153357, 5.95900466730196271449972715732, 6.92116149525582118764831835825, 6.93717508441912222809183241428, 7.17450713167885343099030707177, 7.66138306465432851890178277032, 8.820853343801771743435698779760
