| L(s) = 1 | − 3-s + 5-s − 2·7-s − 2·9-s + 8·11-s − 2·13-s − 15-s − 2·17-s + 10·19-s + 2·21-s + 8·23-s − 6·25-s + 2·27-s + 4·29-s + 11·31-s − 8·33-s − 2·35-s − 2·37-s + 2·39-s − 5·41-s − 5·43-s − 2·45-s + 2·47-s + 3·49-s + 2·51-s + 5·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s + 2.41·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 2.29·19-s + 0.436·21-s + 1.66·23-s − 6/5·25-s + 0.384·27-s + 0.742·29-s + 1.97·31-s − 1.39·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s − 0.780·41-s − 0.762·43-s − 0.298·45-s + 0.291·47-s + 3/7·49-s + 0.280·51-s + 0.686·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556869908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556869908\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11 T + 89 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 7 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 83 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 23 T + 323 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
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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
−11.34248434181394264285703236965, −10.95314083862301695146980952542, −10.08259655619301986781537845446, −9.883127184923822987309935775602, −9.535795387949590173451809107347, −9.084510549202141237522329932422, −8.691166707003106484924568493522, −8.189161116476844874255524753889, −7.31021489287142883396217234915, −7.02891335494972013361125505045, −6.46591444276376579357911502264, −6.30036227906475194920740174462, −5.53163241973076577201627516367, −5.26513122644408668908041073321, −4.48996704181960298666791816275, −3.96049769465594348872907084961, −3.07996379674428868987540109033, −2.92304355581998684891020819472, −1.61734174675707997492615665344, −0.867345877778910985836202396496,
0.867345877778910985836202396496, 1.61734174675707997492615665344, 2.92304355581998684891020819472, 3.07996379674428868987540109033, 3.96049769465594348872907084961, 4.48996704181960298666791816275, 5.26513122644408668908041073321, 5.53163241973076577201627516367, 6.30036227906475194920740174462, 6.46591444276376579357911502264, 7.02891335494972013361125505045, 7.31021489287142883396217234915, 8.189161116476844874255524753889, 8.691166707003106484924568493522, 9.084510549202141237522329932422, 9.535795387949590173451809107347, 9.883127184923822987309935775602, 10.08259655619301986781537845446, 10.95314083862301695146980952542, 11.34248434181394264285703236965
