| L(s) = 1 | + 2-s + 4-s − 5·7-s + 8-s − 5·14-s + 16-s − 15·23-s − 7·25-s − 5·28-s − 2·31-s + 32-s + 12·41-s − 15·46-s − 3·47-s + 7·49-s − 7·50-s − 5·56-s − 2·62-s + 64-s + 9·71-s − 8·73-s − 11·79-s + 12·82-s + 9·89-s − 15·92-s − 3·94-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.88·7-s + 0.353·8-s − 1.33·14-s + 1/4·16-s − 3.12·23-s − 7/5·25-s − 0.944·28-s − 0.359·31-s + 0.176·32-s + 1.87·41-s − 2.21·46-s − 0.437·47-s + 49-s − 0.989·50-s − 0.668·56-s − 0.254·62-s + 1/8·64-s + 1.06·71-s − 0.936·73-s − 1.23·79-s + 1.32·82-s + 0.953·89-s − 1.56·92-s − 0.309·94-s − 1.42·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.652714422744095323466797360572, −9.027659096749185878742715215599, −8.225571133318294688989128787892, −7.81703477466188316464245955479, −7.33220847183364183467575379900, −6.61188979234268611173870798333, −6.13353325696699756481986466494, −5.96267241280273743869494244056, −5.36003727661739443711228678585, −4.25409564691026793235370079339, −3.99502876875398050428326696477, −3.40242420560698287174088785935, −2.63835594809174172822601552921, −1.90599422533204285651276085720, 0,
1.90599422533204285651276085720, 2.63835594809174172822601552921, 3.40242420560698287174088785935, 3.99502876875398050428326696477, 4.25409564691026793235370079339, 5.36003727661739443711228678585, 5.96267241280273743869494244056, 6.13353325696699756481986466494, 6.61188979234268611173870798333, 7.33220847183364183467575379900, 7.81703477466188316464245955479, 8.225571133318294688989128787892, 9.027659096749185878742715215599, 9.652714422744095323466797360572
