| L(s) = 1 | + 3-s + 5-s + 2·7-s − 4·9-s + 2·11-s + 2·13-s + 15-s − 9·17-s + 3·19-s + 2·21-s − 2·23-s − 8·25-s − 6·27-s + 8·29-s − 10·31-s + 2·33-s + 2·35-s + 10·37-s + 2·39-s − 6·41-s − 3·43-s − 4·45-s − 14·47-s + 3·49-s − 9·51-s + 3·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 4/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s − 2.18·17-s + 0.688·19-s + 0.436·21-s − 0.417·23-s − 8/5·25-s − 1.15·27-s + 1.48·29-s − 1.79·31-s + 0.348·33-s + 0.338·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 0.457·43-s − 0.596·45-s − 2.04·47-s + 3/7·49-s − 1.26·51-s + 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64128064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64128064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 87 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 19 T + 201 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15 T + 159 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 53 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 167 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 35 T + 483 T^{2} + 35 p T^{3} + p^{2} T^{4} \) |
| 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
−7.86979118746675367616008922768, −7.38402519896094312906547170505, −6.88437962990276544470056364993, −6.62121698165762204313898167512, −6.14434232743377580946934071034, −6.12620167658657621088943810951, −5.43845426593996724509623397475, −5.41682923818915039535062334747, −4.77859311317401351444478816147, −4.48771422911637910128307028216, −4.02925362213184971309925727782, −3.78040444640529188868115167518, −3.17687928372804894313245555221, −2.93460515195291804292902759477, −2.23829548766659474054653923799, −2.23782016415593851525141109657, −1.50521944352161166703004554658, −1.30180891903276023177608621822, 0, 0,
1.30180891903276023177608621822, 1.50521944352161166703004554658, 2.23782016415593851525141109657, 2.23829548766659474054653923799, 2.93460515195291804292902759477, 3.17687928372804894313245555221, 3.78040444640529188868115167518, 4.02925362213184971309925727782, 4.48771422911637910128307028216, 4.77859311317401351444478816147, 5.41682923818915039535062334747, 5.43845426593996724509623397475, 6.12620167658657621088943810951, 6.14434232743377580946934071034, 6.62121698165762204313898167512, 6.88437962990276544470056364993, 7.38402519896094312906547170505, 7.86979118746675367616008922768
