| L(s) = 1 | − 8·7-s + 9-s − 6·17-s − 8·23-s + 8·31-s + 10·41-s − 16·47-s + 34·49-s − 8·63-s − 16·71-s + 14·73-s + 8·79-s − 8·81-s + 2·89-s − 4·97-s − 16·103-s − 30·113-s + 48·119-s − 15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 1/3·9-s − 1.45·17-s − 1.66·23-s + 1.43·31-s + 1.56·41-s − 2.33·47-s + 34/7·49-s − 1.00·63-s − 1.89·71-s + 1.63·73-s + 0.900·79-s − 8/9·81-s + 0.211·89-s − 0.406·97-s − 1.57·103-s − 2.82·113-s + 4.40·119-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.80066511339698868144867013998, −7.51327785746854395546202593783, −6.83470490320537926686206465359, −6.64524812620345641513724650200, −6.60574079033666831796661054267, −6.24992205017648395450833929133, −5.72544329304210989125572679481, −5.68462424216386743750398770678, −4.90559165246597932010046809390, −4.39857743985684237770994546865, −4.20957376125427192556908452528, −3.82442928585005421839859896951, −3.20576986097163585444709038369, −3.18381671333882572040613513724, −2.48801286959059943482269220320, −2.35344242108488063020296406629, −1.61132926727736960221333292394, −0.846486907140068184631275542643, 0, 0,
0.846486907140068184631275542643, 1.61132926727736960221333292394, 2.35344242108488063020296406629, 2.48801286959059943482269220320, 3.18381671333882572040613513724, 3.20576986097163585444709038369, 3.82442928585005421839859896951, 4.20957376125427192556908452528, 4.39857743985684237770994546865, 4.90559165246597932010046809390, 5.68462424216386743750398770678, 5.72544329304210989125572679481, 6.24992205017648395450833929133, 6.60574079033666831796661054267, 6.64524812620345641513724650200, 6.83470490320537926686206465359, 7.51327785746854395546202593783, 7.80066511339698868144867013998
