| L(s) = 1 | − 2·5-s + 2·13-s − 10·17-s − 25-s − 14·37-s − 20·41-s + 10·53-s + 24·61-s − 4·65-s + 22·73-s + 20·85-s − 26·97-s + 40·101-s − 30·113-s + 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s − 2.42·17-s − 1/5·25-s − 2.30·37-s − 3.12·41-s + 1.37·53-s + 3.07·61-s − 0.496·65-s + 2.57·73-s + 2.16·85-s − 2.63·97-s + 3.98·101-s − 2.82·113-s + 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8848304450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8848304450\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.69070048906599955411700646339, −10.08844439675758383541538758720, −10.01494075156774709424674860623, −9.155922408831269568252927866396, −8.780335037974028390004830251118, −8.480562004187249511904055025426, −8.256940296972529450224820556715, −7.60198276477324176105330257627, −6.91800161053221953059438822094, −6.65049688413170539207160463909, −6.63553547403116105017187727342, −5.46976334895499297820517657150, −5.31758959749386313518572698943, −4.63988777221551882064743399918, −4.09557281354080656170450891442, −3.67102638766190293415429729947, −3.23630675826133321837138035102, −2.18720762862321371885776946629, −1.86923672087254418009846353794, −0.47540063476749427317373765702,
0.47540063476749427317373765702, 1.86923672087254418009846353794, 2.18720762862321371885776946629, 3.23630675826133321837138035102, 3.67102638766190293415429729947, 4.09557281354080656170450891442, 4.63988777221551882064743399918, 5.31758959749386313518572698943, 5.46976334895499297820517657150, 6.63553547403116105017187727342, 6.65049688413170539207160463909, 6.91800161053221953059438822094, 7.60198276477324176105330257627, 8.256940296972529450224820556715, 8.480562004187249511904055025426, 8.780335037974028390004830251118, 9.155922408831269568252927866396, 10.01494075156774709424674860623, 10.08844439675758383541538758720, 10.69070048906599955411700646339
