| L(s) = 1 | + 3-s + 5-s − 2·7-s − 4·9-s + 2·11-s − 2·13-s + 15-s − 5·17-s + 3·19-s − 2·21-s − 4·23-s − 8·25-s − 6·27-s + 2·33-s − 2·35-s + 10·37-s − 2·39-s − 10·41-s − 3·43-s − 4·45-s − 8·47-s + 3·49-s − 5·51-s + 13·53-s + 2·55-s + 3·57-s + 12·59-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 − 1.21·17-s + 0.688·19-s − 0.436·21-s − 0.834·23-s − 8/5·25-s − 1.15·27-s + 0.348·33-s − 0.338·35-s + 1.64·37-s − 0.320·39-s − 1.56·41-s − 0.457·43-s − 0.596·45-s − 1.16·47-s + 3/7·49-s − 0.700·51-s + 1.78·53-s + 0.269·55-s + 0.397·57-s + 1.56·59-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
| \(L(1)\) |
\(\approx\) |
\(2.648693942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648693942\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 + 5 T + 29 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 62 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 57 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 147 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 167 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 21 T + 243 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 167 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 169 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 19 T + 245 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 187 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.966619582513237075922645392857, −7.962012910633720953218213045880, −7.21519805821616025312639012810, −6.86845906974759804657973504486, −6.66528475089868918943734070071, −6.33579525590797103410232898866, −5.94011693254101691768791419811, −5.60999900109687516492151414278, −5.15151557811363224353330832781, −5.12496525060139994173417368556, −4.33057952175874938587359541844, −3.97206369200633276632331992072, −3.56318852774364704102060607471, −3.47333591434763346922513216324, −2.79781088439245053555135025834, −2.38807905729854354043968971114, −2.04566338285766587448203693664, −1.94136736697565553688181183716, −0.73081687646427890641279274082, −0.49213820577522629710346314376,
0.49213820577522629710346314376, 0.73081687646427890641279274082, 1.94136736697565553688181183716, 2.04566338285766587448203693664, 2.38807905729854354043968971114, 2.79781088439245053555135025834, 3.47333591434763346922513216324, 3.56318852774364704102060607471, 3.97206369200633276632331992072, 4.33057952175874938587359541844, 5.12496525060139994173417368556, 5.15151557811363224353330832781, 5.60999900109687516492151414278, 5.94011693254101691768791419811, 6.33579525590797103410232898866, 6.66528475089868918943734070071, 6.86845906974759804657973504486, 7.21519805821616025312639012810, 7.962012910633720953218213045880, 7.966619582513237075922645392857
