| L(s) = 1 | + 262·9-s − 792·11-s − 6.38e3·19-s − 852·29-s + 6.55e3·31-s + 2.49e4·41-s + 5.42e4·49-s − 7.00e4·59-s − 4.82e4·61-s + 1.76e5·71-s + 1.85e5·79-s + 5.32e4·81-s − 3.45e5·89-s − 2.07e5·99-s − 2.39e5·101-s + 5.93e5·109-s − 1.85e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.56e6·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s − 1.97·11-s − 4.05·19-s − 0.188·29-s + 1.22·31-s + 2.31·41-s + 3.22·49-s − 2.62·59-s − 1.66·61-s + 4.14·71-s + 3.35·79-s + 0.902·81-s − 4.62·89-s − 2.12·99-s − 2.33·101-s + 4.78·109-s − 1.15·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4.21·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.240033535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.240033535\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 262 T^{2} + 15367 T^{4} - 562004 p^{2} T^{6} + 15367 p^{10} T^{8} - 262 p^{20} T^{10} + p^{30} T^{12} \) |
| 7 | \( 1 - 54270 T^{2} + 1636355967 T^{4} - 32963917489060 T^{6} + 1636355967 p^{10} T^{8} - 54270 p^{20} T^{10} + p^{30} T^{12} \) |
| 11 | \( ( 1 + 36 p T + 327873 T^{2} + 67617992 T^{3} + 327873 p^{5} T^{4} + 36 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| 13 | \( 1 - 1565442 T^{2} + 1135881445383 T^{4} - 513409425866197436 T^{6} + 1135881445383 p^{10} T^{8} - 1565442 p^{20} T^{10} + p^{30} T^{12} \) |
| 17 | \( 1 - 2487258 T^{2} + 7838169507183 T^{4} - 10425763805475099564 T^{6} + 7838169507183 p^{10} T^{8} - 2487258 p^{20} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 + 168 p T + 9330057 T^{2} + 15933739216 T^{3} + 9330057 p^{5} T^{4} + 168 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 - 21917310 T^{2} + 235143882878367 T^{4} - \)\(17\!\cdots\!80\)\( T^{6} + 235143882878367 p^{10} T^{8} - 21917310 p^{20} T^{10} + p^{30} T^{12} \) |
| 29 | \( ( 1 + 426 T - 939693 T^{2} - 141773503364 T^{3} - 939693 p^{5} T^{4} + 426 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( ( 1 - 3276 T + 2292813 T^{2} + 41639420248 T^{3} + 2292813 p^{5} T^{4} - 3276 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( 1 - 47567538 T^{2} + 8632350495280023 T^{4} - \)\(24\!\cdots\!84\)\( T^{6} + 8632350495280023 p^{10} T^{8} - 47567538 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( ( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 384843783 p^{5} T^{4} - 12450 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 43 | \( 1 - 587098422 T^{2} + 171971726662843287 T^{4} - \)\(31\!\cdots\!16\)\( T^{6} + 171971726662843287 p^{10} T^{8} - 587098422 p^{20} T^{10} + p^{30} T^{12} \) |
| 47 | \( 1 - 888472110 T^{2} + 408520605892907727 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + 408520605892907727 p^{10} T^{8} - 888472110 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 - 343748754 T^{2} + 524992621000918647 T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + 524992621000918647 p^{10} T^{8} - 343748754 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( ( 1 + 35040 T + 1757220897 T^{2} + 50989349593920 T^{3} + 1757220897 p^{5} T^{4} + 35040 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 61 | \( ( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 393086643 p^{5} T^{4} + 24138 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 - 1227086118 T^{2} + 624926915335274247 T^{4} + \)\(11\!\cdots\!36\)\( T^{6} + 624926915335274247 p^{10} T^{8} - 1227086118 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( ( 1 - 88092 T + 7289446053 T^{2} - 329541325840584 T^{3} + 7289446053 p^{5} T^{4} - 88092 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 73 | \( 1 - 3294592458 T^{2} + 16216281639521153535 T^{4} - \)\(29\!\cdots\!40\)\( T^{6} + 16216281639521153535 p^{10} T^{8} - 3294592458 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 - 92952 T + 11164448877 T^{2} - 573846024396496 T^{3} + 11164448877 p^{5} T^{4} - 92952 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 7671858246 T^{2} + 26481453986477119527 T^{4} - \)\(57\!\cdots\!28\)\( T^{6} + 26481453986477119527 p^{10} T^{8} - 7671858246 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( ( 1 + 172686 T + 26445328791 T^{2} + 2103593815517412 T^{3} + 26445328791 p^{5} T^{4} + 172686 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 97 | \( 1 - 31357705242 T^{2} + \)\(46\!\cdots\!35\)\( T^{4} - \)\(45\!\cdots\!60\)\( T^{6} + \)\(46\!\cdots\!35\)\( p^{10} T^{8} - 31357705242 p^{20} T^{10} + p^{30} T^{12} \) |
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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
−4.51060036525045126122463754611, −4.28819707550506685502226607654, −4.27510700780328466384411962874, −4.20950809463450620457655041234, −4.18973034651744355138241214951, −4.04266959197035143768955934205, −3.81304071756519267492800065342, −3.35621724918361390605958825375, −3.08225085141907975634844560266, −2.97867775608266770256789890457, −2.97731168492113993591196069098, −2.94128563501460717487056915411, −2.38229386557624791218506177154, −2.30541374757734125350585263866, −2.08428722872341190305178525225, −1.98273540301333650808755081011, −1.98246307849699444231325905485, −1.75483849318306169772232031473, −1.48290644269186080890234875254, −0.884653843208188171427550138838, −0.855816929614853290730009370386, −0.819621445657402378527360713812, −0.61331162634915997586957487636, −0.25435980667397859355865276757, −0.20014804276314120799015974411,
0.20014804276314120799015974411, 0.25435980667397859355865276757, 0.61331162634915997586957487636, 0.819621445657402378527360713812, 0.855816929614853290730009370386, 0.884653843208188171427550138838, 1.48290644269186080890234875254, 1.75483849318306169772232031473, 1.98246307849699444231325905485, 1.98273540301333650808755081011, 2.08428722872341190305178525225, 2.30541374757734125350585263866, 2.38229386557624791218506177154, 2.94128563501460717487056915411, 2.97731168492113993591196069098, 2.97867775608266770256789890457, 3.08225085141907975634844560266, 3.35621724918361390605958825375, 3.81304071756519267492800065342, 4.04266959197035143768955934205, 4.18973034651744355138241214951, 4.20950809463450620457655041234, 4.27510700780328466384411962874, 4.28819707550506685502226607654, 4.51060036525045126122463754611
Plot not available for L-functions of degree greater than 10.
