| L(s) = 1 | − 4-s − 4·5-s − 8·11-s + 16-s + 6·19-s + 4·20-s + 11·25-s − 4·31-s − 14·41-s + 8·44-s + 13·49-s + 32·55-s − 2·59-s − 20·61-s − 64-s + 24·71-s − 6·76-s − 24·79-s − 4·80-s − 28·89-s − 24·95-s − 11·100-s − 28·101-s − 8·109-s + 26·121-s + 4·124-s − 24·125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 2.41·11-s + 1/4·16-s + 1.37·19-s + 0.894·20-s + 11/5·25-s − 0.718·31-s − 2.18·41-s + 1.20·44-s + 13/7·49-s + 4.31·55-s − 0.260·59-s − 2.56·61-s − 1/8·64-s + 2.84·71-s − 0.688·76-s − 2.70·79-s − 0.447·80-s − 2.96·89-s − 2.46·95-s − 1.09·100-s − 2.78·101-s − 0.766·109-s + 2.36·121-s + 0.359·124-s − 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2780207487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2780207487\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−10.58403663407299744604591365246, −10.03549797516700905913802709913, −9.702542737649110577965230494307, −9.136140051891708449603278342373, −8.581656691902899961938734523949, −8.126952477769167909476606973081, −8.102727965081464282891326794943, −7.50339636247024450277429750782, −7.13873020870527675492195224376, −6.92209984283866388744867712968, −5.86711800645502012029788744544, −5.45054736440685869843757009601, −5.08464785451077947716366161758, −4.68735542222912872036896440229, −4.04188058044116100586125562276, −3.59436265437240855196350297783, −2.92492443369844846466972344573, −2.71073725200747259002831385588, −1.43599999120860242284636575188, −0.27240090702338449410073082496,
0.27240090702338449410073082496, 1.43599999120860242284636575188, 2.71073725200747259002831385588, 2.92492443369844846466972344573, 3.59436265437240855196350297783, 4.04188058044116100586125562276, 4.68735542222912872036896440229, 5.08464785451077947716366161758, 5.45054736440685869843757009601, 5.86711800645502012029788744544, 6.92209984283866388744867712968, 7.13873020870527675492195224376, 7.50339636247024450277429750782, 8.102727965081464282891326794943, 8.126952477769167909476606973081, 8.581656691902899961938734523949, 9.136140051891708449603278342373, 9.702542737649110577965230494307, 10.03549797516700905913802709913, 10.58403663407299744604591365246
