| L(s) = 1 | − 6·3-s − 6·5-s + 10·7-s + 27·9-s − 16·11-s − 26·13-s + 36·15-s + 4·17-s − 70·19-s − 60·21-s + 128·23-s − 110·25-s − 108·27-s + 80·29-s + 250·31-s + 96·33-s − 60·35-s + 152·37-s + 156·39-s − 146·41-s − 504·43-s − 162·45-s + 524·47-s − 498·49-s − 24·51-s + 52·53-s + 96·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.536·5-s + 0.539·7-s + 9-s − 0.438·11-s − 0.554·13-s + 0.619·15-s + 0.0570·17-s − 0.845·19-s − 0.623·21-s + 1.16·23-s − 0.879·25-s − 0.769·27-s + 0.512·29-s + 1.44·31-s + 0.506·33-s − 0.289·35-s + 0.675·37-s + 0.640·39-s − 0.556·41-s − 1.78·43-s − 0.536·45-s + 1.62·47-s − 1.45·49-s − 0.0658·51-s + 0.134·53-s + 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 10 T + 598 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 16 T + 918 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 70 T + 12118 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 80 T + 46310 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 250 T + 49782 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 152 T + 70470 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 146 T + 137634 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 504 T + 215286 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 524 T + 272222 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 52 T + 291198 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 164 T + 406182 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 304 T + 237958 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 914 T + 804838 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 455470 T^{2} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 825950 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 824 T + 1009374 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 828 T + 1032470 T^{2} - 828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 826 T + 1571354 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 552 T + 219630 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.274090572441093308961868206413, −7.972556742047539197724692053360, −7.60692503212540092891439426325, −7.22744247796560373984866853908, −6.81036324951670558010687840380, −6.41211384218241928822686503231, −6.10500889564802934571012423275, −5.66825137743847929935455953648, −5.03646704181157927918863703045, −4.94699665447484958439676613116, −4.39812110862154928310069838310, −4.32279236506706821650487693490, −3.33861678365140594515102483672, −3.28707266826865341493779101878, −2.30859194913056508998304192021, −2.14336848176103515447395174572, −1.22193273094220616114550804154, −0.949913636232847777923886681029, 0, 0,
0.949913636232847777923886681029, 1.22193273094220616114550804154, 2.14336848176103515447395174572, 2.30859194913056508998304192021, 3.28707266826865341493779101878, 3.33861678365140594515102483672, 4.32279236506706821650487693490, 4.39812110862154928310069838310, 4.94699665447484958439676613116, 5.03646704181157927918863703045, 5.66825137743847929935455953648, 6.10500889564802934571012423275, 6.41211384218241928822686503231, 6.81036324951670558010687840380, 7.22744247796560373984866853908, 7.60692503212540092891439426325, 7.972556742047539197724692053360, 8.274090572441093308961868206413
