| L(s) = 1 | − 4·3-s − 3·4-s − 6·5-s + 6·9-s + 12·12-s + 2·13-s + 24·15-s + 5·16-s − 8·19-s + 18·20-s + 17·25-s + 4·27-s − 18·36-s − 8·39-s + 10·41-s − 36·45-s + 8·47-s − 20·48-s − 13·49-s − 6·52-s + 32·57-s − 72·60-s − 61-s − 3·64-s − 12·65-s − 22·73-s − 68·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3/2·4-s − 2.68·5-s + 2·9-s + 3.46·12-s + 0.554·13-s + 6.19·15-s + 5/4·16-s − 1.83·19-s + 4.02·20-s + 17/5·25-s + 0.769·27-s − 3·36-s − 1.28·39-s + 1.56·41-s − 5.36·45-s + 1.16·47-s − 2.88·48-s − 1.85·49-s − 0.832·52-s + 4.23·57-s − 9.29·60-s − 0.128·61-s − 3/8·64-s − 1.48·65-s − 2.57·73-s − 7.85·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226981 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226981 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 61 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.287570693733701613432166498970, −8.274264971280189488991068896861, −7.66039763333771123227596439320, −6.99406793450997328346612383814, −6.57824201628261131943206641630, −5.92888111121729653830785512767, −5.60157612088078313975836309422, −4.88190631622273510627461766193, −4.41578079951852361479456728016, −4.25666915587159511841100490901, −3.77261031711318121333074155223, −2.92919017445927982588513856374, −0.989611421396249014674132130630, 0, 0,
0.989611421396249014674132130630, 2.92919017445927982588513856374, 3.77261031711318121333074155223, 4.25666915587159511841100490901, 4.41578079951852361479456728016, 4.88190631622273510627461766193, 5.60157612088078313975836309422, 5.92888111121729653830785512767, 6.57824201628261131943206641630, 6.99406793450997328346612383814, 7.66039763333771123227596439320, 8.274264971280189488991068896861, 8.287570693733701613432166498970
