| L(s) = 1 | + 128·4-s − 1.76e3·7-s − 1.26e4·13-s + 2.87e4·19-s + 7.81e4·25-s − 2.25e5·28-s − 1.78e5·31-s − 1.23e6·37-s − 1.03e6·43-s + 8.23e5·49-s − 1.61e6·52-s − 1.53e6·61-s − 2.09e6·64-s + 4.05e6·67-s + 2.47e6·73-s + 3.67e6·76-s + 4.24e6·79-s + 2.22e7·91-s − 5.27e6·97-s + 1.00e7·100-s + 2.19e7·103-s − 3.36e7·109-s + 1.94e7·121-s − 2.29e7·124-s + 127-s + 131-s − 5.06e7·133-s + ⋯ |
| L(s) = 1 | + 4-s − 1.94·7-s − 1.59·13-s + 0.960·19-s + 25-s − 1.94·28-s − 1.07·31-s − 3.99·37-s − 1.98·43-s + 49-s − 1.59·52-s − 0.867·61-s − 64-s + 1.64·67-s + 0.744·73-s + 0.960·76-s + 0.968·79-s + 3.09·91-s − 0.586·97-s + 100-s + 1.97·103-s − 2.48·109-s + 121-s − 1.07·124-s − 1.86·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3164412517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3164412517\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 508 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2009 T + p^{7} T^{2} )( 1 + 14614 T + p^{7} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 14357 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 152471 T + p^{7} T^{2} )( 1 + 331387 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 615373 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 409495 T + p^{7} T^{2} )( 1 + 625729 T + p^{7} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1998347 T + p^{7} T^{2} )( 1 + 3535546 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4443527 T + p^{7} T^{2} )( 1 + 385072 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1236809 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8763044 T + p^{7} T^{2} )( 1 + 4517617 T + p^{7} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 + 17521555 T + p^{7} T^{2} ) \) |
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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
−13.07857520438373923668323509377, −12.44347144753196451718539013977, −12.28297793347449692626001099366, −11.69778709026514979993280761315, −10.99544463407257033313709632357, −10.15919742384712090213539427459, −10.15621396212794201208617892235, −9.285611456906382501216314120083, −8.950244195216344250981714363624, −7.901588492854894173373305963443, −7.12847392268299697358099191621, −6.77679090591027005771582096318, −6.53824844798646355358145685476, −5.34867335346696535616141597916, −5.02386681361590739073254449294, −3.51989589663641092076620618782, −3.25103718282067086768544514242, −2.43152442024036846578739891675, −1.61552683602046791291982804596, −0.17302871399128428759465496997,
0.17302871399128428759465496997, 1.61552683602046791291982804596, 2.43152442024036846578739891675, 3.25103718282067086768544514242, 3.51989589663641092076620618782, 5.02386681361590739073254449294, 5.34867335346696535616141597916, 6.53824844798646355358145685476, 6.77679090591027005771582096318, 7.12847392268299697358099191621, 7.901588492854894173373305963443, 8.950244195216344250981714363624, 9.285611456906382501216314120083, 10.15621396212794201208617892235, 10.15919742384712090213539427459, 10.99544463407257033313709632357, 11.69778709026514979993280761315, 12.28297793347449692626001099366, 12.44347144753196451718539013977, 13.07857520438373923668323509377
