| L(s) = 1 | − 2·5-s + 2·7-s + 4·11-s − 14·23-s + 3·25-s − 4·29-s − 4·31-s − 4·35-s + 4·37-s − 16·41-s − 2·43-s − 10·47-s − 6·49-s − 8·53-s − 8·55-s + 8·59-s + 12·61-s + 2·67-s + 4·71-s + 8·77-s − 8·79-s − 6·83-s + 12·89-s − 24·97-s − 20·101-s − 2·103-s + 30·107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.20·11-s − 2.91·23-s + 3/5·25-s − 0.742·29-s − 0.718·31-s − 0.676·35-s + 0.657·37-s − 2.49·41-s − 0.304·43-s − 1.45·47-s − 6/7·49-s − 1.09·53-s − 1.07·55-s + 1.04·59-s + 1.53·61-s + 0.244·67-s + 0.474·71-s + 0.911·77-s − 0.900·79-s − 0.658·83-s + 1.27·89-s − 2.43·97-s − 1.99·101-s − 0.197·103-s + 2.90·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14 T + 90 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 318 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.979951457616915621316638572785, −7.78555646597524839745621593208, −7.17573972772169468807605845322, −6.90454252288475744842854755284, −6.53419375976268307989004205714, −6.30461913471336846363742573750, −5.68361383826243964419202001116, −5.52092631979600732196849015498, −4.81225703263067062477311869741, −4.79761146919639043338462479593, −4.02804029000450458145323669287, −3.98763449331395125518079665087, −3.53997198094196516242735352429, −3.28277747006493414092543848331, −2.44990122470733355634317573783, −2.02920102627528953034464000263, −1.55250921386301456412136359783, −1.25151692496450074590689870294, 0, 0,
1.25151692496450074590689870294, 1.55250921386301456412136359783, 2.02920102627528953034464000263, 2.44990122470733355634317573783, 3.28277747006493414092543848331, 3.53997198094196516242735352429, 3.98763449331395125518079665087, 4.02804029000450458145323669287, 4.79761146919639043338462479593, 4.81225703263067062477311869741, 5.52092631979600732196849015498, 5.68361383826243964419202001116, 6.30461913471336846363742573750, 6.53419375976268307989004205714, 6.90454252288475744842854755284, 7.17573972772169468807605845322, 7.78555646597524839745621593208, 7.979951457616915621316638572785
