| L(s) = 1 | + 2·2-s + 2·4-s − 6·9-s − 2·11-s − 4·16-s + 2·17-s − 12·18-s + 4·19-s − 4·22-s + 6·25-s − 8·32-s + 4·34-s − 12·36-s + 8·38-s − 6·41-s − 4·43-s − 4·44-s + 11·49-s + 12·50-s − 6·59-s − 8·64-s + 14·67-s + 4·68-s − 6·73-s + 8·76-s + 27·81-s − 12·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 2·9-s − 0.603·11-s − 16-s + 0.485·17-s − 2.82·18-s + 0.917·19-s − 0.852·22-s + 6/5·25-s − 1.41·32-s + 0.685·34-s − 2·36-s + 1.29·38-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 11/7·49-s + 1.69·50-s − 0.781·59-s − 64-s + 1.71·67-s + 0.485·68-s − 0.702·73-s + 0.917·76-s + 3·81-s − 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2238016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2238016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.46828759290698556556077222458, −6.76875151668636688190671927234, −6.61714696112762425798852507107, −6.01118959988592874072183945220, −5.55808919246910946026742413690, −5.42343612788189796607524603056, −4.93367392585237899551830209948, −4.66759655092524780233827617395, −3.74129083235295889863164852178, −3.50251652351210255345821016437, −3.01847605105214453787857599755, −2.60812448202309704136989954701, −2.17623176042938741952187369651, −1.01964758259084386618903124498, 0,
1.01964758259084386618903124498, 2.17623176042938741952187369651, 2.60812448202309704136989954701, 3.01847605105214453787857599755, 3.50251652351210255345821016437, 3.74129083235295889863164852178, 4.66759655092524780233827617395, 4.93367392585237899551830209948, 5.42343612788189796607524603056, 5.55808919246910946026742413690, 6.01118959988592874072183945220, 6.61714696112762425798852507107, 6.76875151668636688190671927234, 7.46828759290698556556077222458
