| L(s) = 1 | − 2·3-s + 2·9-s + 12·11-s − 8·23-s + 8·25-s − 6·27-s − 24·33-s − 16·37-s − 32·47-s + 16·49-s + 4·59-s + 16·61-s + 16·69-s − 8·71-s − 8·73-s − 16·75-s + 11·81-s + 20·83-s − 16·97-s + 24·99-s − 12·107-s − 16·109-s + 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s + 3.61·11-s − 1.66·23-s + 8/5·25-s − 1.15·27-s − 4.17·33-s − 2.63·37-s − 4.66·47-s + 16/7·49-s + 0.520·59-s + 2.04·61-s + 1.92·69-s − 0.949·71-s − 0.936·73-s − 1.84·75-s + 11/9·81-s + 2.19·83-s − 1.62·97-s + 2.41·99-s − 1.16·107-s − 1.53·109-s + 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.160166407138318214300651917701, −8.036084077565844897833599344961, −7.85984494523370398543395636615, −7.18525770703991326931854786251, −7.11598315438957270778129682342, −6.79698648521432253349227155231, −6.61433965144405487198754339378, −6.55095056031657267483702077158, −6.40602062376300254341970292162, −5.93826331989077339506237106707, −5.70087326337340059556178010635, −5.38207263894208148420383547763, −5.11666821562140884642663997519, −4.98790155860994296026882195668, −4.38230861728745604736417430792, −4.26104731231808745258099638198, −3.86457554916008562357222693846, −3.74678272242361289903181249756, −3.52229369124652651480767571762, −3.03819750642008649545686305237, −2.52866925710650984195584796654, −1.83548621488853150549222289154, −1.51012220631758179412520442508, −1.45472451603873987349616811025, −0.52833814300405234554427011740,
0.52833814300405234554427011740, 1.45472451603873987349616811025, 1.51012220631758179412520442508, 1.83548621488853150549222289154, 2.52866925710650984195584796654, 3.03819750642008649545686305237, 3.52229369124652651480767571762, 3.74678272242361289903181249756, 3.86457554916008562357222693846, 4.26104731231808745258099638198, 4.38230861728745604736417430792, 4.98790155860994296026882195668, 5.11666821562140884642663997519, 5.38207263894208148420383547763, 5.70087326337340059556178010635, 5.93826331989077339506237106707, 6.40602062376300254341970292162, 6.55095056031657267483702077158, 6.61433965144405487198754339378, 6.79698648521432253349227155231, 7.11598315438957270778129682342, 7.18525770703991326931854786251, 7.85984494523370398543395636615, 8.036084077565844897833599344961, 8.160166407138318214300651917701
