| L(s) = 1 | + 2-s − 2·4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s + 2·11-s − 13-s − 2·14-s + 16-s + 11·17-s + 3·19-s + 4·20-s + 2·22-s + 11·23-s + 3·25-s − 26-s + 4·28-s − 29-s + 6·31-s + 2·32-s + 11·34-s + 4·35-s − 6·37-s + 3·38-s + 6·40-s + 41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s + 0.603·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 2.66·17-s + 0.688·19-s + 0.894·20-s + 0.426·22-s + 2.29·23-s + 3/5·25-s − 0.196·26-s + 0.755·28-s − 0.185·29-s + 1.07·31-s + 0.353·32-s + 1.88·34-s + 0.676·35-s − 0.986·37-s + 0.486·38-s + 0.948·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827868624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827868624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 47 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - T + 81 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 20 T + 181 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 65 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 89 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 161 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 175 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 239 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−10.11368122482799054800079482927, −10.03633210691854259383451190882, −9.298443919758692309828566255004, −9.083105148706136042416193519449, −8.562730392232975027846106217379, −8.309739996522110565456817230949, −7.67395611027058447743680314646, −7.20486014744947811110534132345, −6.97548809816371143076764884443, −6.41657940988785949531951197400, −5.67458381309765849341976589501, −5.35533968460624360786544167907, −4.96734930604886184472534409739, −4.63245396726622019814592377168, −3.69806748059809158939309358786, −3.59487385431958100572888589503, −3.29458834584304556554449057791, −2.61771849054135880875153288362, −1.21663621455259393726745348875, −0.69145868477614976384217980372,
0.69145868477614976384217980372, 1.21663621455259393726745348875, 2.61771849054135880875153288362, 3.29458834584304556554449057791, 3.59487385431958100572888589503, 3.69806748059809158939309358786, 4.63245396726622019814592377168, 4.96734930604886184472534409739, 5.35533968460624360786544167907, 5.67458381309765849341976589501, 6.41657940988785949531951197400, 6.97548809816371143076764884443, 7.20486014744947811110534132345, 7.67395611027058447743680314646, 8.309739996522110565456817230949, 8.562730392232975027846106217379, 9.083105148706136042416193519449, 9.298443919758692309828566255004, 10.03633210691854259383451190882, 10.11368122482799054800079482927
