| L(s) = 1 | + 10·9-s − 2·25-s + 44·29-s + 32·41-s − 6·49-s − 40·61-s + 37·81-s − 72·89-s − 24·101-s + 84·109-s − 42·121-s − 32·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 10/3·9-s − 2/5·25-s + 8.17·29-s + 4.99·41-s − 6/7·49-s − 5.12·61-s + 37/9·81-s − 7.63·89-s − 2.38·101-s + 8.04·109-s − 3.81·121-s − 2.86·125-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(113.6500790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(113.6500790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} + 16 T^{3} + p T^{4} + p^{3} T^{6} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{6} \) |
| good | 3 | \( ( 1 - 5 T^{2} + 19 T^{4} - 74 T^{6} + 19 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 21 T^{2} + 427 T^{4} + 5118 T^{6} + 427 p^{2} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - p T^{2} + 211 T^{4} - 5370 T^{6} + 211 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 7 T^{2} + 531 T^{4} + 5658 T^{6} + 531 p^{2} T^{8} + 7 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} + 219 T^{4} + 10644 T^{6} + 219 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 82 T^{2} + 3439 T^{4} - 94748 T^{6} + 3439 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 11 T + 103 T^{2} - 626 T^{3} + 103 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 31 | \( ( 1 + 58 T^{2} + 2959 T^{4} + 101868 T^{6} + 2959 p^{2} T^{8} + 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 54 T^{2} + 3751 T^{4} - 112244 T^{6} + 3751 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 8 T + 119 T^{2} - 608 T^{3} + 119 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 43 | \( ( 1 - 142 T^{2} + 9831 T^{4} - 479108 T^{6} + 9831 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 25 T^{2} + 6499 T^{4} - 105926 T^{6} + 6499 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 114 T^{2} + 7063 T^{4} - 404124 T^{6} + 7063 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 250 T^{2} + 30379 T^{4} + 2250804 T^{6} + 30379 p^{2} T^{8} + 250 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 10 T + 185 T^{2} + 1096 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 206 T^{2} + 21975 T^{4} - 1631172 T^{6} + 21975 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 322 T^{2} + 47759 T^{4} + 4243772 T^{6} + 47759 p^{2} T^{8} + 322 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 234 T^{2} + 391 p T^{4} - 2449484 T^{6} + 391 p^{3} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 325 T^{2} + 47179 T^{4} + 4383246 T^{6} + 47179 p^{2} T^{8} + 325 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 126 T^{2} + 13915 T^{4} - 1297532 T^{6} + 13915 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 6 T + p T^{2} )^{12} \) |
| 97 | \( ( 1 - 393 T^{2} + 68467 T^{4} - 7712006 T^{6} + 68467 p^{2} T^{8} - 393 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.84282438854785332686379965178, −2.80822870775679608000159198358, −2.60631874975498016254656693364, −2.56298615490994706120610519344, −2.49394194373985360154089748798, −2.30808724706677274581712361477, −2.29575559977214116769719190059, −2.23592569627127486867300547767, −1.92866225287688273440355583566, −1.87878628048414817571971004181, −1.83553166227328797718539638765, −1.71691289004744151308542182241, −1.53628408277633602534198719791, −1.50744115460158168926659031559, −1.37638405053788023582895289753, −1.36788012108938129339666527421, −1.20198387760098223908542902484, −1.17119617808064601953412422157, −1.14875821880919529245146554179, −0.932428178719764353349741626252, −0.61165325579151750415129617715, −0.59156304385804532084819310439, −0.49037125656644304994425319159, −0.44368318902205629116568611012, −0.40752295417172116218769531437,
0.40752295417172116218769531437, 0.44368318902205629116568611012, 0.49037125656644304994425319159, 0.59156304385804532084819310439, 0.61165325579151750415129617715, 0.932428178719764353349741626252, 1.14875821880919529245146554179, 1.17119617808064601953412422157, 1.20198387760098223908542902484, 1.36788012108938129339666527421, 1.37638405053788023582895289753, 1.50744115460158168926659031559, 1.53628408277633602534198719791, 1.71691289004744151308542182241, 1.83553166227328797718539638765, 1.87878628048414817571971004181, 1.92866225287688273440355583566, 2.23592569627127486867300547767, 2.29575559977214116769719190059, 2.30808724706677274581712361477, 2.49394194373985360154089748798, 2.56298615490994706120610519344, 2.60631874975498016254656693364, 2.80822870775679608000159198358, 2.84282438854785332686379965178
Plot not available for L-functions of degree greater than 10.
