| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 6·7-s − 4·8-s − 4·10-s − 8·11-s − 4·13-s + 12·14-s + 8·16-s − 2·17-s + 2·19-s + 4·20-s + 16·22-s + 3·25-s + 8·26-s − 12·28-s − 4·29-s − 6·31-s − 8·32-s + 4·34-s − 12·35-s − 2·37-s − 4·38-s − 8·40-s − 4·41-s − 10·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 2.26·7-s − 1.41·8-s − 1.26·10-s − 2.41·11-s − 1.10·13-s + 3.20·14-s + 2·16-s − 0.485·17-s + 0.458·19-s + 0.894·20-s + 3.41·22-s + 3/5·25-s + 1.56·26-s − 2.26·28-s − 0.742·29-s − 1.07·31-s − 1.41·32-s + 0.685·34-s − 2.02·35-s − 0.328·37-s − 0.648·38-s − 1.26·40-s − 0.624·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 128 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 215 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 143 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 120 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.53395455354729580116053469386, −10.47901817533946089705980451277, −9.891659987097994186587409226244, −9.722832552934395336803449497777, −9.222697266578735014282348495178, −9.160920829589346233820277538329, −8.113279120425039489990623479181, −8.103368373131588726861552509454, −7.31996317957000879743322008349, −6.77562535273111037724955155460, −6.50951253662799041173106328615, −5.84110997200682682606964101848, −5.30600954979009837211133848160, −5.05003835867806828695676183738, −3.61864026751503349114017566624, −2.99718508586738244117959337162, −2.75549156342552231674726398882, −1.94056688123354518976348832487, 0, 0,
1.94056688123354518976348832487, 2.75549156342552231674726398882, 2.99718508586738244117959337162, 3.61864026751503349114017566624, 5.05003835867806828695676183738, 5.30600954979009837211133848160, 5.84110997200682682606964101848, 6.50951253662799041173106328615, 6.77562535273111037724955155460, 7.31996317957000879743322008349, 8.103368373131588726861552509454, 8.113279120425039489990623479181, 9.160920829589346233820277538329, 9.222697266578735014282348495178, 9.722832552934395336803449497777, 9.891659987097994186587409226244, 10.47901817533946089705980451277, 10.53395455354729580116053469386
