| L(s) = 1 | + 4·2-s + 3·3-s + 12·4-s + 10·5-s + 12·6-s + 12·7-s + 32·8-s − 29·9-s + 40·10-s − 12·11-s + 36·12-s − 51·13-s + 48·14-s + 30·15-s + 80·16-s − 66·17-s − 116·18-s − 16·19-s + 120·20-s + 36·21-s − 48·22-s + 46·23-s + 96·24-s − 102·25-s − 204·26-s − 120·27-s + 144·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s + 0.647·7-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 + 0.916·14-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 + 0.374·21-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 + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.539234498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.539234498\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 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 | $D_{4}$ | \( 1 - 2 p T + 202 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 12 T + 430 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−15.18647883234539920959157725650, −14.83062085865878235674905538912, −14.24910721329052471017921724141, −14.03975996965972779158563317700, −13.25584438413473762133121447209, −13.06172084985184722101079549290, −12.04862656122644248621567956276, −11.73676836993597692492847111683, −10.83719144434063244108004924646, −10.54321426930488727621815369202, −9.340141671499293458159369275635, −9.006143647598720824849157842862, −7.80439388841851506960715737271, −7.46350918882600668900206455713, −6.20594417356575346379787881362, −5.75926941555892148818995126679, −4.94466632988669100584039586253, −4.10727139240804765228356108711, −2.69823751029223412056299479304, −2.22804957238931827498818513754,
2.22804957238931827498818513754, 2.69823751029223412056299479304, 4.10727139240804765228356108711, 4.94466632988669100584039586253, 5.75926941555892148818995126679, 6.20594417356575346379787881362, 7.46350918882600668900206455713, 7.80439388841851506960715737271, 9.006143647598720824849157842862, 9.340141671499293458159369275635, 10.54321426930488727621815369202, 10.83719144434063244108004924646, 11.73676836993597692492847111683, 12.04862656122644248621567956276, 13.06172084985184722101079549290, 13.25584438413473762133121447209, 14.03975996965972779158563317700, 14.24910721329052471017921724141, 14.83062085865878235674905538912, 15.18647883234539920959157725650
