| L(s) = 1 | − 8·11-s − 12·17-s + 4·19-s − 2·25-s + 20·41-s − 12·43-s + 4·49-s + 8·67-s + 32·73-s − 32·83-s − 28·89-s − 8·97-s − 8·107-s + 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 2.91·17-s + 0.917·19-s − 2/5·25-s + 3.12·41-s − 1.82·43-s + 4/7·49-s + 0.977·67-s + 3.74·73-s − 3.51·83-s − 2.96·89-s − 0.812·97-s − 0.773·107-s + 1.12·113-s + 2.36·121-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 − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−7.47353079413488525209888820142, −7.35229165952742296717957220045, −6.76618706210449371393566242939, −6.65268754008567685500474121345, −6.23782918670541635752755051796, −5.74335450187239271949866898163, −5.30708158582515008666792853379, −5.29806027342339766692163331735, −4.82990973098821535575196630441, −4.39913015718352033740208682832, −4.02972088139529867561238918084, −3.82925724862281323159012983789, −3.04047482774543277921856084358, −2.61951661971750158076718846391, −2.55001344315535648888936639273, −2.17755764266242356530247112161, −1.55994931356633437273324915499, −0.872977023725410083820691039316, 0, 0,
0.872977023725410083820691039316, 1.55994931356633437273324915499, 2.17755764266242356530247112161, 2.55001344315535648888936639273, 2.61951661971750158076718846391, 3.04047482774543277921856084358, 3.82925724862281323159012983789, 4.02972088139529867561238918084, 4.39913015718352033740208682832, 4.82990973098821535575196630441, 5.29806027342339766692163331735, 5.30708158582515008666792853379, 5.74335450187239271949866898163, 6.23782918670541635752755051796, 6.65268754008567685500474121345, 6.76618706210449371393566242939, 7.35229165952742296717957220045, 7.47353079413488525209888820142
