| L(s) = 1 | + 6·3-s + 6·5-s + 21·9-s + 36·15-s − 4·17-s + 6·19-s + 21·25-s + 56·27-s + 12·31-s + 126·45-s + 14·49-s − 24·51-s + 36·57-s − 16·59-s − 16·61-s + 20·67-s + 16·71-s − 8·73-s + 126·75-s − 8·79-s + 126·81-s − 24·85-s + 72·93-s + 36·95-s − 12·101-s − 32·103-s + 16·107-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 2.68·5-s + 7·9-s + 9.29·15-s − 0.970·17-s + 1.37·19-s + 21/5·25-s + 10.7·27-s + 2.15·31-s + 18.7·45-s + 2·49-s − 3.36·51-s + 4.76·57-s − 2.08·59-s − 2.04·61-s + 2.44·67-s + 1.89·71-s − 0.936·73-s + 14.5·75-s − 0.900·79-s + 14·81-s − 2.60·85-s + 7.46·93-s + 3.69·95-s − 1.19·101-s − 3.15·103-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(157.5296057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(157.5296057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( ( 1 - T )^{6} \) |
| 19 | \( 1 - 6 T + 13 T^{2} + 4 T^{3} + 13 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 2 p T^{2} + 85 T^{4} - 456 T^{6} + 85 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 4 p T^{2} + 909 T^{4} - 12000 T^{6} + 909 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 50 T^{2} + 1213 T^{4} - 18984 T^{6} + 1213 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T + 5 T^{2} - 88 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 104 T^{2} + 5127 T^{4} - 149376 T^{6} + 5127 p^{2} T^{8} - 104 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 8 T^{2} + 2445 T^{4} - 12864 T^{6} + 2445 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 6 T + 91 T^{2} - 368 T^{3} + 91 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 170 T^{2} + 13405 T^{4} - 626088 T^{6} + 13405 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 64 T^{2} + 5029 T^{4} - 197664 T^{6} + 5029 p^{2} T^{8} - 64 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 206 T^{2} + 19357 T^{4} - 1060200 T^{6} + 19357 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 40 T^{2} - 1481 T^{4} + 135936 T^{6} - 1481 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 284 T^{2} + 35247 T^{4} - 2437656 T^{6} + 35247 p^{2} T^{8} - 284 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 73 T^{2} + 176 T^{3} + 73 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 8 T + 25 T^{2} + 288 T^{3} + 25 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 10 T + 169 T^{2} - 1212 T^{3} + 169 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 8 T + 109 T^{2} - 368 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 4 T + 159 T^{2} + 552 T^{3} + 159 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 4 T + 217 T^{2} + 600 T^{3} + 217 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 224 T^{2} + 33231 T^{4} - 3281616 T^{6} + 33231 p^{2} T^{8} - 224 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 80 T^{2} + 11157 T^{4} - 1499232 T^{6} + 11157 p^{2} T^{8} - 80 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 410 T^{2} + 76261 T^{4} - 8900712 T^{6} + 76261 p^{2} T^{8} - 410 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.29545759190883916924411178965, −4.11966391760457706667860301044, −3.93877105368231593638522753713, −3.66439711839121187226220305010, −3.63507533309830281167738766545, −3.43906416586518157199255090182, −3.43213815892485423887811106463, −3.12812196362361423609160358934, −2.97425888457489037795315764132, −2.90170614978822426749132787852, −2.88488958536124095851427459413, −2.64945514652933965653788654693, −2.57528048158978341751358870108, −2.33078958863853214793551429698, −2.17462958026142884452628282929, −2.14309881235108311472015986476, −1.97097929146070074038713796714, −1.72901787235794688296472763455, −1.65144556158986412850285318518, −1.53854868221792931938840250557, −1.19758518214071097704278029252, −1.06712657597452140813574819594, −0.940474889656043028858291791565, −0.63334779707832422623855757176, −0.40672576343319535643677622047,
0.40672576343319535643677622047, 0.63334779707832422623855757176, 0.940474889656043028858291791565, 1.06712657597452140813574819594, 1.19758518214071097704278029252, 1.53854868221792931938840250557, 1.65144556158986412850285318518, 1.72901787235794688296472763455, 1.97097929146070074038713796714, 2.14309881235108311472015986476, 2.17462958026142884452628282929, 2.33078958863853214793551429698, 2.57528048158978341751358870108, 2.64945514652933965653788654693, 2.88488958536124095851427459413, 2.90170614978822426749132787852, 2.97425888457489037795315764132, 3.12812196362361423609160358934, 3.43213815892485423887811106463, 3.43906416586518157199255090182, 3.63507533309830281167738766545, 3.66439711839121187226220305010, 3.93877105368231593638522753713, 4.11966391760457706667860301044, 4.29545759190883916924411178965
Plot not available for L-functions of degree greater than 10.
