L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·9-s − 3·11-s + 5·16-s + 4·18-s − 6·22-s − 6·23-s − 8·25-s + 6·32-s + 6·36-s − 5·37-s − 9·43-s − 9·44-s − 12·46-s − 16·50-s − 9·53-s + 7·64-s − 7·67-s + 3·71-s + 8·72-s − 10·74-s − 24·79-s − 5·81-s − 18·86-s − 12·88-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2/3·9-s − 0.904·11-s + 5/4·16-s + 0.942·18-s − 1.27·22-s − 1.25·23-s − 8/5·25-s + 1.06·32-s + 36-s − 0.821·37-s − 1.37·43-s − 1.35·44-s − 1.76·46-s − 2.26·50-s − 1.23·53-s + 7/8·64-s − 0.855·67-s + 0.356·71-s + 0.942·72-s − 1.16·74-s − 2.70·79-s − 5/9·81-s − 1.94·86-s − 1.27·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
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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
−7.66109140875782292797439734506, −7.38022491709403490834438782513, −6.82325337565643340745723175151, −6.39702282875957859237193755899, −5.98600293927349867048524374338, −5.44179789280298378384022589030, −5.25712198345805760925672550290, −4.53946070710420453213655625238, −4.20599149051304473265306609294, −3.79497807382036326947700718378, −3.14617807019897249368473777505, −2.72126381842469168923278952382, −1.85505636399294786038277846477, −1.64121900689994963251181644061, 0,
1.64121900689994963251181644061, 1.85505636399294786038277846477, 2.72126381842469168923278952382, 3.14617807019897249368473777505, 3.79497807382036326947700718378, 4.20599149051304473265306609294, 4.53946070710420453213655625238, 5.25712198345805760925672550290, 5.44179789280298378384022589030, 5.98600293927349867048524374338, 6.39702282875957859237193755899, 6.82325337565643340745723175151, 7.38022491709403490834438782513, 7.66109140875782292797439734506