| L(s) = 1 | + 4·2-s − 3·3-s + 12·4-s − 10·5-s − 12·6-s + 32·8-s − 29·9-s − 40·10-s − 12·11-s − 36·12-s + 51·13-s + 30·15-s + 80·16-s + 66·17-s − 116·18-s + 16·19-s − 120·20-s − 48·22-s + 46·23-s − 96·24-s − 102·25-s + 204·26-s + 120·27-s − 57·29-s + 120·30-s + 17·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s + 1.41·8-s − 1.07·9-s − 1.26·10-s − 0.328·11-s − 0.866·12-s + 1.08·13-s + 0.516·15-s + 5/4·16-s + 0.941·17-s − 1.51·18-s + 0.193·19-s − 1.34·20-s − 0.465·22-s + 0.417·23-s − 0.816·24-s − 0.815·25-s + 1.53·26-s + 0.855·27-s − 0.364·29-s + 0.730·30-s + 0.0984·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5080516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5080516 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 38 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 202 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 51 T + 2836 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 66 T + 10258 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16 T + 6482 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 57 T + 29716 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T - 27234 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 206 T + 85562 T^{2} - 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 373 T + 3836 p T^{2} + 373 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 314 T + 60950 T^{2} - 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 859 T + 380710 T^{2} + 859 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 50 T + 272026 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 612 T + 438694 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1062 T + 674530 T^{2} - 1062 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 844 T + 722378 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 399 T + 747574 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1277 T + 1185552 T^{2} - 1277 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 122 T + 977462 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1802 T + 1946542 T^{2} + 1802 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2046 T + 2450554 T^{2} + 2046 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 910 T + 662234 T^{2} - 910 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.162889009552602294181453330613, −8.090536348939550719534202170578, −7.76188840686237989086919319308, −7.23751385200936398460061222504, −6.59626593570861254644306540182, −6.57579829511820589130392500583, −5.97672561447676353025157537566, −5.54919293971300598348733980328, −5.44465835973939616448873339390, −4.96798745089323424220283297200, −4.45672545316847206766154441900, −3.98215591874131931031467672338, −3.48453480355801118441674635196, −3.41980830657220066875069720155, −2.69906245147283147191011501672, −2.40673171734908573687266624004, −1.36458130653509153350932514836, −1.21576741934918205902719968133, 0, 0,
1.21576741934918205902719968133, 1.36458130653509153350932514836, 2.40673171734908573687266624004, 2.69906245147283147191011501672, 3.41980830657220066875069720155, 3.48453480355801118441674635196, 3.98215591874131931031467672338, 4.45672545316847206766154441900, 4.96798745089323424220283297200, 5.44465835973939616448873339390, 5.54919293971300598348733980328, 5.97672561447676353025157537566, 6.57579829511820589130392500583, 6.59626593570861254644306540182, 7.23751385200936398460061222504, 7.76188840686237989086919319308, 8.090536348939550719534202170578, 8.162889009552602294181453330613
