| L(s) = 1 | − 48·7-s + 38·9-s + 100·17-s + 112·23-s + 246·25-s − 320·31-s + 396·41-s + 1.05e3·47-s + 1.04e3·49-s − 1.82e3·63-s − 1.45e3·71-s − 308·73-s − 1.31e3·79-s + 715·81-s − 1.42e3·89-s − 956·97-s + 1.93e3·103-s − 1.88e3·113-s − 4.80e3·119-s + 726·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3.80e3·153-s + ⋯ |
| L(s) = 1 | − 2.59·7-s + 1.40·9-s + 1.42·17-s + 1.01·23-s + 1.96·25-s − 1.85·31-s + 1.50·41-s + 3.27·47-s + 3.03·49-s − 3.64·63-s − 2.43·71-s − 0.493·73-s − 1.86·79-s + 0.980·81-s − 1.70·89-s − 1.00·97-s + 1.85·103-s − 1.56·113-s − 3.69·119-s + 6/11·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 2.00·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.904572024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904572024\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 246 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 p^{2} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11782 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9574 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75062 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 528 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 239190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 35466 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 151462 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 728 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 154 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 656 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1087878 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 714 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 478 T + p^{3} T^{2} )^{2} \) |
| 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
−12.11612707463161875523685853223, −11.22764714299319912895341675107, −10.57723738449281527308274480460, −10.42538900663903027115445844205, −9.820017050472189887352634511801, −9.550780307567013017507173911884, −8.886174068167655909965859509063, −8.847881655349570779554331129719, −7.48749552587012996451964213837, −7.24457423617757938796820382894, −7.05410609786882479615949649298, −6.28950060603509690117392218994, −5.81526475236632263694665973477, −5.28002614422809057896061428657, −4.26080451446738959082772263379, −3.87861137604062335193575549335, −2.98632984558379587973162317561, −2.86153153938665675655990324674, −1.35557532313524436370529560537, −0.59803423341275942997376131769,
0.59803423341275942997376131769, 1.35557532313524436370529560537, 2.86153153938665675655990324674, 2.98632984558379587973162317561, 3.87861137604062335193575549335, 4.26080451446738959082772263379, 5.28002614422809057896061428657, 5.81526475236632263694665973477, 6.28950060603509690117392218994, 7.05410609786882479615949649298, 7.24457423617757938796820382894, 7.48749552587012996451964213837, 8.847881655349570779554331129719, 8.886174068167655909965859509063, 9.550780307567013017507173911884, 9.820017050472189887352634511801, 10.42538900663903027115445844205, 10.57723738449281527308274480460, 11.22764714299319912895341675107, 12.11612707463161875523685853223
