| L(s) = 1 | − 4·25-s − 20·49-s + 56·73-s − 8·97-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4/5·25-s − 2.85·49-s + 6.55·73-s − 0.812·97-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8496216425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8496216425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 38 T^{2} + 483 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 94 T^{2} + 6027 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2}( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.96035286650184237615237025471, −3.80917812705153792661962208206, −3.74248239984377486702722548093, −3.69678586003659980215384721049, −3.26161796332726943390088839197, −3.16516056386780110122146432996, −3.15716282426807416775720311507, −3.13298603663865883417467757297, −3.03881267763968231959156789020, −2.99119108868141739971339530666, −2.50248292156764671185475834325, −2.36755874411560810190576121228, −2.35627210649878045731503736545, −2.31433029288576652432558176454, −1.97670375471459973610302788714, −1.95706488907124557596769140871, −1.89403003864042369036349197633, −1.67513433721757761762131185169, −1.45503478619317998939426802400, −1.12788810739386009682150581441, −1.04979217871838222188891443650, −1.04222930162486571314382971479, −0.63187572186566204374713609688, −0.43157326477089477572046485786, −0.096981138953401852999747934456,
0.096981138953401852999747934456, 0.43157326477089477572046485786, 0.63187572186566204374713609688, 1.04222930162486571314382971479, 1.04979217871838222188891443650, 1.12788810739386009682150581441, 1.45503478619317998939426802400, 1.67513433721757761762131185169, 1.89403003864042369036349197633, 1.95706488907124557596769140871, 1.97670375471459973610302788714, 2.31433029288576652432558176454, 2.35627210649878045731503736545, 2.36755874411560810190576121228, 2.50248292156764671185475834325, 2.99119108868141739971339530666, 3.03881267763968231959156789020, 3.13298603663865883417467757297, 3.15716282426807416775720311507, 3.16516056386780110122146432996, 3.26161796332726943390088839197, 3.69678586003659980215384721049, 3.74248239984377486702722548093, 3.80917812705153792661962208206, 3.96035286650184237615237025471
Plot not available for L-functions of degree greater than 10.
