| L(s) = 1 | + 114·7-s + 4.02e3·13-s − 1.85e3·19-s + 5.07e4·25-s − 708·31-s − 6.61e4·37-s + 1.16e5·43-s − 2.27e5·49-s + 1.70e5·61-s + 8.41e4·67-s + 7.48e5·73-s + 5.60e5·79-s + 4.58e5·91-s + 1.96e6·97-s + 4.12e6·103-s + 5.67e6·109-s + 7.06e6·121-s + 127-s + 131-s − 2.11e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.332·7-s + 1.83·13-s − 0.270·19-s + 3.24·25-s − 0.0237·31-s − 1.30·37-s + 1.46·43-s − 1.93·49-s + 0.749·61-s + 0.279·67-s + 1.92·73-s + 1.13·79-s + 0.609·91-s + 2.15·97-s + 3.77·103-s + 4.38·109-s + 3.98·121-s − 0.0898·133-s + 0.697·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+3)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(51.65727770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(51.65727770\) |
| \(L(4)\) |
|
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 | \( 1 - 50718 T^{2} + 60204279 p^{2} T^{4} - 45452366628 p^{4} T^{6} + 60204279 p^{14} T^{8} - 50718 p^{24} T^{10} + p^{36} T^{12} \) |
| 7 | \( ( 1 - 57 T + 118830 T^{2} - 5686557 T^{3} + 118830 p^{6} T^{4} - 57 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 11 | \( 1 - 7066206 T^{2} + 206017614615 p^{2} T^{4} - 3761639607943332 p^{4} T^{6} + 206017614615 p^{14} T^{8} - 7066206 p^{24} T^{10} + p^{36} T^{12} \) |
| 13 | \( ( 1 - 2013 T + 4393902 T^{2} - 8896652809 T^{3} + 4393902 p^{6} T^{4} - 2013 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 17 | \( 1 - 53129838 T^{2} + 1502011683819183 T^{4} - \)\(10\!\cdots\!24\)\( p^{2} T^{6} + 1502011683819183 p^{12} T^{8} - 53129838 p^{24} T^{10} + p^{36} T^{12} \) |
| 19 | \( ( 1 + 927 T + 97462998 T^{2} + 185727048923 T^{3} + 97462998 p^{6} T^{4} + 927 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 23 | \( 1 - 224272014 T^{2} - 15274701790383345 T^{4} + \)\(79\!\cdots\!72\)\( T^{6} - 15274701790383345 p^{12} T^{8} - 224272014 p^{24} T^{10} + p^{36} T^{12} \) |
| 29 | \( 1 - 356667702 T^{2} + 639250131336555039 T^{4} - \)\(31\!\cdots\!04\)\( T^{6} + 639250131336555039 p^{12} T^{8} - 356667702 p^{24} T^{10} + p^{36} T^{12} \) |
| 31 | \( ( 1 + 354 T + 2300139759 T^{2} + 3240452278748 T^{3} + 2300139759 p^{6} T^{4} + 354 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 37 | \( ( 1 + 33099 T + 2496495342 T^{2} - 40424677692657 T^{3} + 2496495342 p^{6} T^{4} + 33099 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 41 | \( 1 - 19129636806 T^{2} + \)\(17\!\cdots\!55\)\( T^{4} - \)\(97\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!55\)\( p^{12} T^{8} - 19129636806 p^{24} T^{10} + p^{36} T^{12} \) |
| 43 | \( ( 1 - 58086 T + 13398041799 T^{2} - 597218629282868 T^{3} + 13398041799 p^{6} T^{4} - 58086 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 47 | \( 1 - 38545955502 T^{2} + \)\(78\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{6} + \)\(78\!\cdots\!59\)\( p^{12} T^{8} - 38545955502 p^{24} T^{10} + p^{36} T^{12} \) |
| 53 | \( 1 - 63806695062 T^{2} + \)\(15\!\cdots\!71\)\( T^{4} - \)\(28\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!71\)\( p^{12} T^{8} - 63806695062 p^{24} T^{10} + p^{36} T^{12} \) |
| 59 | \( 1 - 184309841502 T^{2} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(88\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!55\)\( p^{12} T^{8} - 184309841502 p^{24} T^{10} + p^{36} T^{12} \) |
| 61 | \( ( 1 - 85077 T + 53361734238 T^{2} - 4230858455153521 T^{3} + 53361734238 p^{6} T^{4} - 85077 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 67 | \( ( 1 - 42081 T + 83616104694 T^{2} + 15350350935302939 T^{3} + 83616104694 p^{6} T^{4} - 42081 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 71 | \( 1 + 48259046682 T^{2} + \)\(26\!\cdots\!51\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{6} + \)\(26\!\cdots\!51\)\( p^{12} T^{8} + 48259046682 p^{24} T^{10} + p^{36} T^{12} \) |
| 73 | \( ( 1 - 374493 T + 483983490966 T^{2} - 114035013331623673 T^{3} + 483983490966 p^{6} T^{4} - 374493 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 79 | \( ( 1 - 280065 T + 238124449614 T^{2} + 8347989726521323 T^{3} + 238124449614 p^{6} T^{4} - 280065 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
| 83 | \( 1 - 489108707958 T^{2} + \)\(18\!\cdots\!67\)\( T^{4} - \)\(64\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!67\)\( p^{12} T^{8} - 489108707958 p^{24} T^{10} + p^{36} T^{12} \) |
| 89 | \( 1 - 1170039211854 T^{2} + \)\(73\!\cdots\!71\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{6} + \)\(73\!\cdots\!71\)\( p^{12} T^{8} - 1170039211854 p^{24} T^{10} + p^{36} T^{12} \) |
| 97 | \( ( 1 - 983709 T + 2704711329414 T^{2} - 1651129803516612953 T^{3} + 2704711329414 p^{6} T^{4} - 983709 p^{12} T^{5} + p^{18} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.18377711363104612027809946697, −4.59319281097237334549126532829, −4.52804501170799067565671021220, −4.39176450791148638491549110786, −4.35147334609541803756894372808, −4.29753765862217870408766055413, −3.95980702228042692526267022798, −3.39782208183138601870857663881, −3.29403880095381462311242485361, −3.26374875804022618723447232691, −3.22977744528940387990062252037, −3.15102795121350780083222265599, −2.99478924019312381849056023001, −2.27732495074358195905196619056, −2.24859012660361927321536602663, −1.93156962591593821897099509387, −1.90129386554793897218451313488, −1.84015158839849737416083878432, −1.59272847220922050426491483259, −1.06682928268017276838750145518, −0.863942934815386067723440056113, −0.74930939662222679467003090022, −0.60547883655965315283086522729, −0.55996478060542790839169621379, −0.43989642027876736358662370034,
0.43989642027876736358662370034, 0.55996478060542790839169621379, 0.60547883655965315283086522729, 0.74930939662222679467003090022, 0.863942934815386067723440056113, 1.06682928268017276838750145518, 1.59272847220922050426491483259, 1.84015158839849737416083878432, 1.90129386554793897218451313488, 1.93156962591593821897099509387, 2.24859012660361927321536602663, 2.27732495074358195905196619056, 2.99478924019312381849056023001, 3.15102795121350780083222265599, 3.22977744528940387990062252037, 3.26374875804022618723447232691, 3.29403880095381462311242485361, 3.39782208183138601870857663881, 3.95980702228042692526267022798, 4.29753765862217870408766055413, 4.35147334609541803756894372808, 4.39176450791148638491549110786, 4.52804501170799067565671021220, 4.59319281097237334549126532829, 5.18377711363104612027809946697
Plot not available for L-functions of degree greater than 10.
