| L(s) = 1 | − 2·4-s − 2·5-s − 4·7-s − 4·9-s − 4·13-s − 6·17-s − 2·19-s + 4·20-s − 6·23-s + 3·25-s + 8·28-s + 6·29-s + 4·31-s + 8·35-s + 8·36-s − 10·37-s − 12·43-s + 8·45-s + 6·47-s − 2·49-s + 8·52-s + 6·59-s − 4·61-s + 16·63-s + 8·64-s + 8·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 1.51·7-s − 4/3·9-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 0.894·20-s − 1.25·23-s + 3/5·25-s + 1.51·28-s + 1.11·29-s + 0.718·31-s + 1.35·35-s + 4/3·36-s − 1.64·37-s − 1.82·43-s + 1.19·45-s + 0.875·47-s − 2/7·49-s + 1.10·52-s + 0.781·59-s − 0.512·61-s + 2.01·63-s + 64-s + 0.992·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 177 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−11.54698164439106226729930783707, −10.75445268759827671922665699530, −10.33605324234639532745155791704, −9.968229113168013042007625257549, −9.311733080854377405199147619434, −9.046741351401829290296455114382, −8.482989751725664910727724266830, −8.262402666869140414768348045000, −7.62785178213003284444661141606, −6.72960553697855713400358106824, −6.62343677114704298548679338638, −6.08557060117450825803808158388, −5.03734422369903349739315657878, −4.95399900448559705863593191245, −4.08664773212077312112885437320, −3.64441970285313080261460564600, −2.90381425217480915630800080863, −2.31619596389586267562516240086, 0, 0,
2.31619596389586267562516240086, 2.90381425217480915630800080863, 3.64441970285313080261460564600, 4.08664773212077312112885437320, 4.95399900448559705863593191245, 5.03734422369903349739315657878, 6.08557060117450825803808158388, 6.62343677114704298548679338638, 6.72960553697855713400358106824, 7.62785178213003284444661141606, 8.262402666869140414768348045000, 8.482989751725664910727724266830, 9.046741351401829290296455114382, 9.311733080854377405199147619434, 9.968229113168013042007625257549, 10.33605324234639532745155791704, 10.75445268759827671922665699530, 11.54698164439106226729930783707
