| L(s) = 1 | − 6·5-s − 2·9-s + 8·13-s − 6·17-s + 17·25-s − 12·29-s − 4·37-s − 12·41-s + 12·45-s − 13·49-s − 24·53-s + 2·61-s − 48·65-s − 14·73-s − 5·81-s + 36·85-s + 24·89-s + 16·97-s − 12·101-s + 32·109-s + 12·113-s − 16·117-s − 13·121-s − 18·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 2/3·9-s + 2.21·13-s − 1.45·17-s + 17/5·25-s − 2.22·29-s − 0.657·37-s − 1.87·41-s + 1.78·45-s − 1.85·49-s − 3.29·53-s + 0.256·61-s − 5.95·65-s − 1.63·73-s − 5/9·81-s + 3.90·85-s + 2.54·89-s + 1.62·97-s − 1.19·101-s + 3.06·109-s + 1.12·113-s − 1.47·117-s − 1.18·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−7.65894969899749366452649043261, −7.15811652768508964807251071197, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −6.05813289271022268292942638231, −4.96903618429312636799150610379, −4.80419327140120764096716079767, −4.19306407725294924458030580762, −3.67310116806022314895429635564, −3.33866387034377157662854676826, −3.31935920587958999413340629739, −2.04259190544357015606449170426, −1.39430917521719853589747801849, 0, 0,
1.39430917521719853589747801849, 2.04259190544357015606449170426, 3.31935920587958999413340629739, 3.33866387034377157662854676826, 3.67310116806022314895429635564, 4.19306407725294924458030580762, 4.80419327140120764096716079767, 4.96903618429312636799150610379, 6.05813289271022268292942638231, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.15811652768508964807251071197, 7.65894969899749366452649043261
