| L(s) = 1 | + 2-s − 2·7-s + 8-s + 2·11-s − 6·13-s − 2·14-s − 16-s + 4·17-s + 2·22-s + 6·23-s − 6·26-s + 10·29-s − 4·31-s − 6·32-s + 4·34-s − 4·37-s − 2·41-s + 6·43-s + 6·46-s + 4·47-s + 2·49-s + 4·53-s − 2·56-s + 10·58-s − 10·59-s + 6·61-s − 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.755·7-s + 0.353·8-s + 0.603·11-s − 1.66·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.426·22-s + 1.25·23-s − 1.17·26-s + 1.85·29-s − 0.718·31-s − 1.06·32-s + 0.685·34-s − 0.657·37-s − 0.312·41-s + 0.914·43-s + 0.884·46-s + 0.583·47-s + 2/7·49-s + 0.549·53-s − 0.267·56-s + 1.31·58-s − 1.30·59-s + 0.768·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257861468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257861468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 209 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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
−10.93900464417342361202518169876, −10.12995998372511462725254404269, −9.791229459639564600931829226461, −9.694663166412464267717980925879, −8.840751624789754143138402320812, −8.793489444800141369825885630290, −8.077064998311500585519750471015, −7.39694107835839205568077898131, −7.10982850448288831188770012406, −6.88911872566188338537162762036, −6.22830252441334005706426709542, −5.66514047862406893472001172422, −5.09159605105069817325765803213, −4.88369549470009831921934399415, −4.21731729233957029325861966132, −3.78192100371466722153694618355, −3.02669625142096373478940055820, −2.68654245032889525692072208461, −1.80480433633238358010960385504, −0.73148826163501750075410840339,
0.73148826163501750075410840339, 1.80480433633238358010960385504, 2.68654245032889525692072208461, 3.02669625142096373478940055820, 3.78192100371466722153694618355, 4.21731729233957029325861966132, 4.88369549470009831921934399415, 5.09159605105069817325765803213, 5.66514047862406893472001172422, 6.22830252441334005706426709542, 6.88911872566188338537162762036, 7.10982850448288831188770012406, 7.39694107835839205568077898131, 8.077064998311500585519750471015, 8.793489444800141369825885630290, 8.840751624789754143138402320812, 9.694663166412464267717980925879, 9.791229459639564600931829226461, 10.12995998372511462725254404269, 10.93900464417342361202518169876
