| L(s) = 1 | − 2·5-s + 2·9-s − 2·11-s − 8·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 4·45-s + 10·49-s + 4·55-s − 8·59-s + 28·61-s + 16·71-s + 16·79-s − 5·81-s + 20·89-s + 16·95-s − 4·99-s − 20·101-s + 28·109-s + 3·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s − 0.603·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s + 0.539·55-s − 1.04·59-s + 3.58·61-s + 1.89·71-s + 1.80·79-s − 5/9·81-s + 2.11·89-s + 1.64·95-s − 0.402·99-s − 1.99·101-s + 2.68·109-s + 3/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163399959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163399959\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.36572114004635848600956117352, −10.78848901929084714073547113363, −10.55787383037823770628685891191, −9.930172879529839923082036369264, −9.810647581571463515380527240658, −9.003076710520285613371303604748, −8.418010201702661654127423564143, −8.169014449639384430285692889352, −7.907089846034882614817261646092, −7.02631659470772875607280564961, −6.87215146085167049520911596624, −6.28286464040817646115025342283, −5.69791440170000227383143639971, −4.85343387815503816826307657787, −4.65709912619538378777439344959, −3.86596933353002954570529155011, −3.60347317798932890194676622399, −2.54521012174282195832422725661, −2.03082661722900013689147938860, −0.68484837455258599651846059410,
0.68484837455258599651846059410, 2.03082661722900013689147938860, 2.54521012174282195832422725661, 3.60347317798932890194676622399, 3.86596933353002954570529155011, 4.65709912619538378777439344959, 4.85343387815503816826307657787, 5.69791440170000227383143639971, 6.28286464040817646115025342283, 6.87215146085167049520911596624, 7.02631659470772875607280564961, 7.907089846034882614817261646092, 8.169014449639384430285692889352, 8.418010201702661654127423564143, 9.003076710520285613371303604748, 9.810647581571463515380527240658, 9.930172879529839923082036369264, 10.55787383037823770628685891191, 10.78848901929084714073547113363, 11.36572114004635848600956117352
