| L(s) = 1 | + 4·5-s − 4·9-s + 4·11-s − 8·17-s − 4·19-s − 2·23-s + 4·25-s − 4·31-s + 4·37-s − 12·41-s + 16·43-s − 16·45-s − 12·47-s + 4·53-s + 16·55-s − 16·59-s + 12·61-s + 12·67-s − 8·71-s − 4·73-s + 4·79-s + 7·81-s − 4·83-s − 32·85-s − 16·89-s − 16·95-s − 16·99-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 4/3·9-s + 1.20·11-s − 1.94·17-s − 0.917·19-s − 0.417·23-s + 4/5·25-s − 0.718·31-s + 0.657·37-s − 1.87·41-s + 2.43·43-s − 2.38·45-s − 1.75·47-s + 0.549·53-s + 2.15·55-s − 2.08·59-s + 1.53·61-s + 1.46·67-s − 0.949·71-s − 0.468·73-s + 0.450·79-s + 7/9·81-s − 0.439·83-s − 3.47·85-s − 1.69·89-s − 1.64·95-s − 1.60·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 128 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 132 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 192 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
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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
−7.60691621055889285999400314869, −6.92526640681535204601198178625, −6.66008065401982429643587860756, −6.64824591810639667729383856926, −6.08279296619129739130607474632, −6.02077556456350389879955155080, −5.51118019023620692117757661132, −5.32384900082566488607404224981, −4.89915319084540518884399473584, −4.31464090091669933490657822344, −4.02318440135931337362989621420, −3.85037123209990006653593276253, −3.04966215125812588893419773538, −2.75017303983036886175526833076, −2.21498743832620310756554232025, −2.18084514948185533162273487238, −1.51161320831097085566708581541, −1.25652271193843151499025816842, 0, 0,
1.25652271193843151499025816842, 1.51161320831097085566708581541, 2.18084514948185533162273487238, 2.21498743832620310756554232025, 2.75017303983036886175526833076, 3.04966215125812588893419773538, 3.85037123209990006653593276253, 4.02318440135931337362989621420, 4.31464090091669933490657822344, 4.89915319084540518884399473584, 5.32384900082566488607404224981, 5.51118019023620692117757661132, 6.02077556456350389879955155080, 6.08279296619129739130607474632, 6.64824591810639667729383856926, 6.66008065401982429643587860756, 6.92526640681535204601198178625, 7.60691621055889285999400314869
