L(s) = 1 | − 2·4-s − 2·9-s + 2·11-s − 3·16-s − 6·19-s + 36·29-s + 4·36-s + 12·41-s − 4·44-s − 23·49-s + 20·59-s − 14·61-s + 8·64-s + 52·71-s + 12·76-s − 24·79-s + 81-s + 24·89-s − 4·99-s + 20·101-s + 40·109-s − 72·116-s − 31·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 4-s − 2/3·9-s + 0.603·11-s − 3/4·16-s − 1.37·19-s + 6.68·29-s + 2/3·36-s + 1.87·41-s − 0.603·44-s − 3.28·49-s + 2.60·59-s − 1.79·61-s + 64-s + 6.17·71-s + 1.37·76-s − 2.70·79-s + 1/9·81-s + 2.54·89-s − 0.402·99-s + 1.99·101-s + 3.83·109-s − 6.68·116-s − 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.511054389\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.511054389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
good | 2 | \( 1 + p T^{2} + 7 T^{4} + 3 p^{2} T^{6} + 7 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 + 2 T^{2} + p T^{4} + 20 T^{6} + p^{3} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 23 T^{2} + 307 T^{4} + 2586 T^{6} + 307 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 50 T^{2} + 1315 T^{4} + 21108 T^{6} + 1315 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 43 T^{2} + 1331 T^{4} + 25042 T^{6} + 1331 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 102 T^{2} + 4831 T^{4} + 138580 T^{6} + 4831 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 166 T^{2} + 13011 T^{4} + 608316 T^{6} + 13011 p^{2} T^{8} + 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 6 T + 79 T^{2} - 516 T^{3} + 79 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 239 T^{2} + 24571 T^{4} + 1388154 T^{6} + 24571 p^{2} T^{8} + 239 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 95 T^{2} + 5443 T^{4} + 214314 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 162 T^{2} + 11539 T^{4} + 610708 T^{6} + 11539 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 10 T + 185 T^{2} - 1132 T^{3} + 185 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 62 T^{2} + 4771 T^{4} + 199788 T^{6} + 4771 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 26 T + 413 T^{2} - 4124 T^{3} + 413 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 307 T^{2} + 43299 T^{4} + 3822498 T^{6} + 43299 p^{2} T^{8} + 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 229 T^{2} + 1864 T^{3} + 229 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 270 T^{2} + 39367 T^{4} + 3817060 T^{6} + 39367 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 12 T - 17 T^{2} + 1320 T^{3} - 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 554 T^{2} + 130507 T^{4} + 16717956 T^{6} + 130507 p^{2} T^{8} + 554 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.04040376453593740695364213529, −5.94699259565334212563688337217, −5.41661366155861086766703204900, −5.39421644333512201886752838619, −5.16646578318485492417535539239, −4.91908694619561831623350060779, −4.81723383056450431016870446997, −4.61746627410580203861480523527, −4.55690299393144658623841334770, −4.44968527760017289098763587493, −4.21181576347591212337222857738, −4.00445050697294118143489854704, −3.74002195520786417313797068315, −3.45307914098692482568328632010, −3.23585636369928182955420610606, −3.20032577494930613732152515152, −2.73889190584271940849123491468, −2.62838815475436065308179961045, −2.49386011183497425266052928076, −2.15062333303143639152485539722, −1.93948977006853465589612621317, −1.57096617156208095214207025588, −0.937803168317371011469960140801, −0.804271158418313148776300199802, −0.55456626669807145697844161379,
0.55456626669807145697844161379, 0.804271158418313148776300199802, 0.937803168317371011469960140801, 1.57096617156208095214207025588, 1.93948977006853465589612621317, 2.15062333303143639152485539722, 2.49386011183497425266052928076, 2.62838815475436065308179961045, 2.73889190584271940849123491468, 3.20032577494930613732152515152, 3.23585636369928182955420610606, 3.45307914098692482568328632010, 3.74002195520786417313797068315, 4.00445050697294118143489854704, 4.21181576347591212337222857738, 4.44968527760017289098763587493, 4.55690299393144658623841334770, 4.61746627410580203861480523527, 4.81723383056450431016870446997, 4.91908694619561831623350060779, 5.16646578318485492417535539239, 5.39421644333512201886752838619, 5.41661366155861086766703204900, 5.94699259565334212563688337217, 6.04040376453593740695364213529
Plot not available for L-functions of degree greater than 10.