| L(s) = 1 | − 3·5-s + 4·7-s − 3·9-s − 3·11-s + 13-s − 3·17-s − 7·19-s − 9·23-s + 5·25-s − 3·29-s − 16·31-s − 12·35-s + 37-s − 3·41-s − 43-s + 9·45-s + 9·49-s − 3·53-s + 9·55-s + 4·61-s − 12·63-s − 3·65-s + 8·67-s − 24·71-s − 11·73-s − 12·77-s + 32·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s − 9-s − 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.60·19-s − 1.87·23-s + 25-s − 0.557·29-s − 2.87·31-s − 2.02·35-s + 0.164·37-s − 0.468·41-s − 0.152·43-s + 1.34·45-s + 9/7·49-s − 0.412·53-s + 1.21·55-s + 0.512·61-s − 1.51·63-s − 0.372·65-s + 0.977·67-s − 2.84·71-s − 1.28·73-s − 1.36·77-s + 3.60·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2710302538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2710302538\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.66254984973676347377975244596, −9.776821956213305281573722737937, −9.135959489140984409522019366506, −8.696300147409355328495147833772, −8.428826621037116129879141103598, −8.223596465920531972536540562352, −7.66592794011883750422174849471, −7.45777023735805085764870625157, −7.01786376633221569298974729601, −6.11132563404614580886402593925, −6.01449311511046754430397384730, −5.30968695071870815697396813611, −4.93586228874475039371567334231, −4.44349045357828938087357071859, −3.80353503678282861715344759396, −3.75210440072080894988617971878, −2.79433002171681351406492356928, −2.04048886502950537780268404559, −1.81813297776412226049428693039, −0.22440494053290833674281439569,
0.22440494053290833674281439569, 1.81813297776412226049428693039, 2.04048886502950537780268404559, 2.79433002171681351406492356928, 3.75210440072080894988617971878, 3.80353503678282861715344759396, 4.44349045357828938087357071859, 4.93586228874475039371567334231, 5.30968695071870815697396813611, 6.01449311511046754430397384730, 6.11132563404614580886402593925, 7.01786376633221569298974729601, 7.45777023735805085764870625157, 7.66592794011883750422174849471, 8.223596465920531972536540562352, 8.428826621037116129879141103598, 8.696300147409355328495147833772, 9.135959489140984409522019366506, 9.776821956213305281573722737937, 10.66254984973676347377975244596
