| L(s) = 1 | − 2·4-s + 8·5-s + 3·16-s − 24·19-s − 16·20-s + 38·25-s − 16·31-s − 8·41-s + 8·49-s − 24·59-s + 48·61-s − 4·64-s + 48·76-s − 16·79-s + 24·80-s + 24·89-s − 192·95-s − 76·100-s + 16·101-s − 16·109-s + 4·121-s + 32·124-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 3.57·5-s + 3/4·16-s − 5.50·19-s − 3.57·20-s + 38/5·25-s − 2.87·31-s − 1.24·41-s + 8/7·49-s − 3.12·59-s + 6.14·61-s − 1/2·64-s + 5.50·76-s − 1.80·79-s + 2.68·80-s + 2.54·89-s − 19.6·95-s − 7.59·100-s + 1.59·101-s − 1.53·109-s + 4/11·121-s + 2.87·124-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.947972532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.947972532\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 442 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 11378 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 4118 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 148 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 11538 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 10578 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.01346689259205706839601122809, −5.98861725166168852055516137501, −5.77877858393814011771884109636, −5.36276285823082138273837300217, −5.31865263381309163936451456359, −5.19534768082190336096555025724, −5.16477346938447626857722608463, −4.61689160070126557307331085074, −4.49037499901271941204917541102, −4.37015708223829015969982634648, −4.21222382493353566844289108862, −3.78909877979420307068281951418, −3.56034688239445638859493665558, −3.53815455958854310181144654529, −3.22733608796866160349886412763, −2.49637824876114744802667501522, −2.47195283835074942003540811684, −2.37741885935425227188931316570, −2.25514342503771385329676757305, −1.99117567434106259796710959050, −1.61636034350377696011495375979, −1.52170477230154553816749219646, −1.25943692057206533591662828052, −0.54480829931739303927939302466, −0.27850336471684107560193235177,
0.27850336471684107560193235177, 0.54480829931739303927939302466, 1.25943692057206533591662828052, 1.52170477230154553816749219646, 1.61636034350377696011495375979, 1.99117567434106259796710959050, 2.25514342503771385329676757305, 2.37741885935425227188931316570, 2.47195283835074942003540811684, 2.49637824876114744802667501522, 3.22733608796866160349886412763, 3.53815455958854310181144654529, 3.56034688239445638859493665558, 3.78909877979420307068281951418, 4.21222382493353566844289108862, 4.37015708223829015969982634648, 4.49037499901271941204917541102, 4.61689160070126557307331085074, 5.16477346938447626857722608463, 5.19534768082190336096555025724, 5.31865263381309163936451456359, 5.36276285823082138273837300217, 5.77877858393814011771884109636, 5.98861725166168852055516137501, 6.01346689259205706839601122809
