| L(s) = 1 | + 3·2-s + 4·4-s + 9·8-s − 9·9-s + 6·11-s + 27·16-s − 27·18-s + 18·22-s − 18·23-s − 25·25-s − 108·29-s + 36·32-s − 36·36-s + 38·37-s + 116·43-s + 24·44-s − 54·46-s − 75·50-s + 6·53-s − 324·58-s + 17·64-s + 118·67-s + 228·71-s − 81·72-s + 114·74-s + 94·79-s + 348·86-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 4-s + 9/8·8-s − 9-s + 6/11·11-s + 1.68·16-s − 3/2·18-s + 9/11·22-s − 0.782·23-s − 25-s − 3.72·29-s + 9/8·32-s − 36-s + 1.02·37-s + 2.69·43-s + 6/11·44-s − 1.17·46-s − 3/2·50-s + 6/53·53-s − 5.58·58-s + 0.265·64-s + 1.76·67-s + 3.21·71-s − 9/8·72-s + 1.54·74-s + 1.18·79-s + 4.04·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.383839113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.383839113\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T - 205 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T + 75 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 2773 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T + 9435 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 114 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T + 2595 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.42754443751509126646191000587, −14.66421272453221876331266098561, −14.62323729447796826303083785051, −13.72610505118807531063048971193, −13.71166887994936687481880775685, −12.72579363999346889019388002482, −12.57037253021287043585273200531, −11.62207360017369720918612416610, −11.17553886714646115839752435728, −10.74861130940744692752212285048, −9.509603005906909382149497478277, −9.297302362551114490375461803652, −7.945552666723230257615597998841, −7.68644464243382614153619475999, −6.57461554296066031581627467039, −5.54855639271986489619732279567, −5.51131183714359580398166671900, −4.01862293385741039919700360561, −3.81225750084565737646213906065, −2.21459161841151590161561784571,
2.21459161841151590161561784571, 3.81225750084565737646213906065, 4.01862293385741039919700360561, 5.51131183714359580398166671900, 5.54855639271986489619732279567, 6.57461554296066031581627467039, 7.68644464243382614153619475999, 7.945552666723230257615597998841, 9.297302362551114490375461803652, 9.509603005906909382149497478277, 10.74861130940744692752212285048, 11.17553886714646115839752435728, 11.62207360017369720918612416610, 12.57037253021287043585273200531, 12.72579363999346889019388002482, 13.71166887994936687481880775685, 13.72610505118807531063048971193, 14.62323729447796826303083785051, 14.66421272453221876331266098561, 15.42754443751509126646191000587
