| L(s) = 1 | + 4-s − 4·9-s + 16-s + 4·17-s − 14·19-s + 2·25-s − 4·36-s + 10·43-s + 6·49-s − 10·53-s + 64-s − 28·67-s + 4·68-s − 14·76-s + 7·81-s − 14·83-s + 2·100-s − 20·101-s + 12·103-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4/3·9-s + 1/4·16-s + 0.970·17-s − 3.21·19-s + 2/5·25-s − 2/3·36-s + 1.52·43-s + 6/7·49-s − 1.37·53-s + 1/8·64-s − 3.42·67-s + 0.485·68-s − 1.60·76-s + 7/9·81-s − 1.53·83-s + 1/5·100-s − 1.99·101-s + 1.18·103-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−8.818001422341887598055025361551, −8.506386751353504503074838176781, −7.909884873535826997331572457762, −7.53772681560611883269190910867, −6.86551951128815142970124111414, −6.32992424983550518816531870956, −5.92492945396306401518910221775, −5.67198169274401755749224526617, −4.75659777205891742843452023924, −4.30041521318027237674925045515, −3.62693366394057502066378982668, −2.78854752382548820150984043245, −2.48198655518567836104940215680, −1.52733658566606030618253294093, 0,
1.52733658566606030618253294093, 2.48198655518567836104940215680, 2.78854752382548820150984043245, 3.62693366394057502066378982668, 4.30041521318027237674925045515, 4.75659777205891742843452023924, 5.67198169274401755749224526617, 5.92492945396306401518910221775, 6.32992424983550518816531870956, 6.86551951128815142970124111414, 7.53772681560611883269190910867, 7.909884873535826997331572457762, 8.506386751353504503074838176781, 8.818001422341887598055025361551
