| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 3·8-s − 9-s + 2·10-s + 11-s + 12-s − 5·13-s − 2·15-s + 16-s + 5·17-s + 18-s + 6·19-s − 2·20-s − 22-s − 2·23-s − 3·24-s + 3·25-s + 5·26-s + 29-s + 2·30-s + 32-s + 33-s − 5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s − 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 1.38·13-s − 0.516·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.37·19-s − 0.447·20-s − 0.213·22-s − 0.417·23-s − 0.612·24-s + 3/5·25-s + 0.980·26-s + 0.185·29-s + 0.365·30-s + 0.176·32-s + 0.174·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8808081666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8808081666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−12.39152531173178272798178925853, −11.65016902251928448488266502048, −11.57622023847228582806782556307, −11.00999536014841663679749943218, −10.18596987098761460739555390847, −9.864532786482428891623367190679, −9.241269512968953197108791280327, −9.199021502286803988284556965835, −8.204534069241880410157454415098, −8.120118791033059228781512183021, −7.38602793913671059379780480666, −7.24028012026210075841974736225, −6.38351817577691066733008491555, −5.75829736128359356907834071908, −5.14643255163729445555918035432, −4.34821352863166038169746696774, −3.59277794653554234167306445068, −2.88370558660059795276118620728, −2.45367621932544607572322756811, −0.837178062567184828536418787995,
0.837178062567184828536418787995, 2.45367621932544607572322756811, 2.88370558660059795276118620728, 3.59277794653554234167306445068, 4.34821352863166038169746696774, 5.14643255163729445555918035432, 5.75829736128359356907834071908, 6.38351817577691066733008491555, 7.24028012026210075841974736225, 7.38602793913671059379780480666, 8.120118791033059228781512183021, 8.204534069241880410157454415098, 9.199021502286803988284556965835, 9.241269512968953197108791280327, 9.864532786482428891623367190679, 10.18596987098761460739555390847, 11.00999536014841663679749943218, 11.57622023847228582806782556307, 11.65016902251928448488266502048, 12.39152531173178272798178925853
