| L(s) = 1 | − 4-s + 2·5-s − 9-s + 16-s + 4·19-s − 2·20-s − 25-s + 16·29-s − 8·31-s + 36-s − 20·41-s − 2·45-s + 24·59-s − 4·61-s − 64-s + 12·71-s − 4·76-s − 8·79-s + 2·80-s + 81-s − 28·89-s + 8·95-s + 100-s + 12·101-s + 4·109-s − 16·116-s − 22·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 1/3·9-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 2.97·29-s − 1.43·31-s + 1/6·36-s − 3.12·41-s − 0.298·45-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 1.42·71-s − 0.458·76-s − 0.900·79-s + 0.223·80-s + 1/9·81-s − 2.96·89-s + 0.820·95-s + 1/10·100-s + 1.19·101-s + 0.383·109-s − 1.48·116-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.085747669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085747669\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.729200764796119309181695624812, −9.353434673950186741501825360433, −8.861454167459039808761036671625, −8.577952003610346816454468393426, −8.105252529058410679088420808538, −7.981977415397311781779893080535, −7.07290660704053498239680768188, −6.78407442677367540423210949802, −6.66147214651667964574699566764, −5.78841169345574841325169294100, −5.64468713834121442748398412585, −5.08203269684506091718479567691, −4.92719653795454403056502388287, −4.17220307708890667225788901902, −3.72258312326547493989756657378, −3.10495908572285584295515131038, −2.74580131967526383533619143421, −1.98376918962552250048975071727, −1.44465338010747811144219153142, −0.60111634040376251457823421806,
0.60111634040376251457823421806, 1.44465338010747811144219153142, 1.98376918962552250048975071727, 2.74580131967526383533619143421, 3.10495908572285584295515131038, 3.72258312326547493989756657378, 4.17220307708890667225788901902, 4.92719653795454403056502388287, 5.08203269684506091718479567691, 5.64468713834121442748398412585, 5.78841169345574841325169294100, 6.66147214651667964574699566764, 6.78407442677367540423210949802, 7.07290660704053498239680768188, 7.981977415397311781779893080535, 8.105252529058410679088420808538, 8.577952003610346816454468393426, 8.861454167459039808761036671625, 9.353434673950186741501825360433, 9.729200764796119309181695624812
