| L(s) = 1 | − 5-s + 2·9-s + 4·11-s + 2·13-s + 4·17-s + 12·19-s − 2·25-s + 4·37-s − 2·45-s + 2·49-s − 4·55-s − 4·59-s − 2·65-s + 4·67-s − 24·79-s − 5·81-s − 12·83-s − 4·85-s − 12·95-s + 8·99-s − 24·103-s − 4·109-s − 4·113-s + 4·117-s − 6·121-s + 10·125-s + 127-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2/3·9-s + 1.20·11-s + 0.554·13-s + 0.970·17-s + 2.75·19-s − 2/5·25-s + 0.657·37-s − 0.298·45-s + 2/7·49-s − 0.539·55-s − 0.520·59-s − 0.248·65-s + 0.488·67-s − 2.70·79-s − 5/9·81-s − 1.31·83-s − 0.433·85-s − 1.23·95-s + 0.804·99-s − 2.36·103-s − 0.383·109-s − 0.376·113-s + 0.369·117-s − 0.545·121-s + 0.894·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.117396641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117396641\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.117421653671586813266405213339, −8.593631180839430835377838256285, −7.966426079446993756000846373600, −7.60623590010481518643358407853, −7.19426798589639996540396316685, −6.77235450050653575886759107610, −6.09878835919971051546496417989, −5.54688175679406757253645675591, −5.22099424574918792536697068747, −4.24177323828206065139649065134, −4.05031802431707437226167480446, −3.29290533862603305121610232408, −2.88997843750678363405458694624, −1.49671912923580168401035638668, −1.10251098034769519741233554296,
1.10251098034769519741233554296, 1.49671912923580168401035638668, 2.88997843750678363405458694624, 3.29290533862603305121610232408, 4.05031802431707437226167480446, 4.24177323828206065139649065134, 5.22099424574918792536697068747, 5.54688175679406757253645675591, 6.09878835919971051546496417989, 6.77235450050653575886759107610, 7.19426798589639996540396316685, 7.60623590010481518643358407853, 7.966426079446993756000846373600, 8.593631180839430835377838256285, 9.117421653671586813266405213339
