| L(s) = 1 | − 2-s + 6·3-s − 11·4-s − 10·5-s − 6·6-s − 4·7-s + 15·8-s + 27·9-s + 10·10-s − 22·11-s − 66·12-s − 90·13-s + 4·14-s − 60·15-s + 61·16-s − 16·17-s − 27·18-s − 170·19-s + 110·20-s − 24·21-s + 22·22-s − 124·23-s + 90·24-s + 75·25-s + 90·26-s + 108·27-s + 44·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s − 1.37·4-s − 0.894·5-s − 0.408·6-s − 0.215·7-s + 0.662·8-s + 9-s + 0.316·10-s − 0.603·11-s − 1.58·12-s − 1.92·13-s + 0.0763·14-s − 1.03·15-s + 0.953·16-s − 0.228·17-s − 0.353·18-s − 2.05·19-s + 1.22·20-s − 0.249·21-s + 0.213·22-s − 1.12·23-s + 0.765·24-s + 3/5·25-s + 0.678·26-s + 0.769·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 622 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 90 T + 6402 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 16 T + 1662 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 170 T + 994 p T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 124 T + 12878 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 158 T + 50106 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 372 T + 82590 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 38 T + 37410 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 p T + 160230 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 224 T + 466 p T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 472 T + 190182 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 248 T + 181334 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 744 T + 738822 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2060 T + 1768494 T^{2} - 2060 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 486 T + 822786 T^{2} + 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 642 T + 691166 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 286 T + 392750 T^{2} + 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 244 T + 1355190 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 168 T + 1053870 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.23792679064640545073851997807, −11.85094862469132100314779326842, −10.90404551148103832380410403804, −10.37622224819429722673356132702, −9.908130172095537346889267006193, −9.597167129720741986027704417040, −8.894333398709967078008516095039, −8.541040759243852194573333948699, −8.072731700173424179396919041230, −7.73987841290595667377532682041, −7.03109962557393122411992906109, −6.46857242033609109549585217379, −5.09468681984002093006600297473, −4.94825489858654283715416737940, −3.96145918542717960996069715914, −3.84880495187668241124743405475, −2.67272850484331780433849265635, −1.95852522855644175105115195687, 0, 0,
1.95852522855644175105115195687, 2.67272850484331780433849265635, 3.84880495187668241124743405475, 3.96145918542717960996069715914, 4.94825489858654283715416737940, 5.09468681984002093006600297473, 6.46857242033609109549585217379, 7.03109962557393122411992906109, 7.73987841290595667377532682041, 8.072731700173424179396919041230, 8.541040759243852194573333948699, 8.894333398709967078008516095039, 9.597167129720741986027704417040, 9.908130172095537346889267006193, 10.37622224819429722673356132702, 10.90404551148103832380410403804, 11.85094862469132100314779326842, 12.23792679064640545073851997807
