| L(s) = 1 | − 2·4-s − 6·5-s − 8·13-s + 4·16-s + 6·17-s + 12·20-s + 17·25-s − 12·29-s + 4·37-s + 12·41-s − 13·49-s + 16·52-s − 24·53-s − 2·61-s − 8·64-s + 48·65-s − 12·68-s − 14·73-s − 24·80-s − 36·85-s − 24·89-s + 16·97-s − 34·100-s − 12·101-s − 32·109-s − 12·113-s + 24·116-s + ⋯ |
| L(s) = 1 | − 4-s − 2.68·5-s − 2.21·13-s + 16-s + 1.45·17-s + 2.68·20-s + 17/5·25-s − 2.22·29-s + 0.657·37-s + 1.87·41-s − 1.85·49-s + 2.21·52-s − 3.29·53-s − 0.256·61-s − 64-s + 5.95·65-s − 1.45·68-s − 1.63·73-s − 2.68·80-s − 3.90·85-s − 2.54·89-s + 1.62·97-s − 3.39·100-s − 1.19·101-s − 3.06·109-s − 1.12·113-s + 2.22·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.968751144610877316457317025190, −7.80441727232688730491090849584, −7.43141403312000422638829977686, −7.09896358802950320449173631503, −6.24695455384273016222733694355, −5.51097905283736056328679977781, −5.16165536636701751832135979028, −4.41466206651142691425988858089, −4.38988833936187387176065759569, −3.73798930153556479954881225054, −3.23857552173567549926694025031, −2.76577588204614961513690244188, −1.40889938157129298285707376171, 0, 0,
1.40889938157129298285707376171, 2.76577588204614961513690244188, 3.23857552173567549926694025031, 3.73798930153556479954881225054, 4.38988833936187387176065759569, 4.41466206651142691425988858089, 5.16165536636701751832135979028, 5.51097905283736056328679977781, 6.24695455384273016222733694355, 7.09896358802950320449173631503, 7.43141403312000422638829977686, 7.80441727232688730491090849584, 7.968751144610877316457317025190
