| L(s) = 1 | − 8·4-s − 20·5-s + 24·16-s + 32·17-s + 160·20-s + 186·25-s − 10·49-s − 18·64-s − 256·68-s − 480·80-s + 56·83-s − 640·85-s − 52·89-s − 1.48e3·100-s + 52·121-s − 1.04e3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4·4-s − 8.94·5-s + 6·16-s + 7.76·17-s + 35.7·20-s + 37.1·25-s − 1.42·49-s − 9/4·64-s − 31.0·68-s − 53.6·80-s + 6.14·83-s − 69.4·85-s − 5.51·89-s − 148.·100-s + 4.72·121-s − 93.3·125-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 + 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06859466726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06859466726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 10 T^{2} + 95 T^{4} + 12 p^{2} T^{6} + 95 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 10 T^{2} + 1151 T^{4} + 12492 T^{6} + 1151 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 2 | \( ( 1 + p^{2} T^{2} + 3 p^{2} T^{4} + 25 T^{6} + 3 p^{4} T^{8} + p^{6} T^{10} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 + p T + 16 T^{2} + 36 T^{3} + 16 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{4} \) |
| 11 | \( ( 1 - 26 T^{2} + 394 T^{4} - 4526 T^{6} + 394 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 5 T^{2} + 150 T^{4} - 332 T^{6} + 150 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 8 T + 60 T^{2} - 258 T^{3} + 60 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 19 | \( ( 1 + 56 T^{2} + 1624 T^{4} + 34970 T^{6} + 1624 p^{2} T^{8} + 56 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 72 T^{2} + 3272 T^{4} + 103898 T^{6} + 3272 p^{2} T^{8} + 72 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 26 T^{2} + 2298 T^{4} - 51734 T^{6} + 2298 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 174 T^{2} + 13874 T^{4} - 650674 T^{6} + 13874 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 198 T^{2} + 17786 T^{4} - 929722 T^{6} + 17786 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + T^{2} + 4110 T^{4} + 21652 T^{6} + 4110 p^{2} T^{8} + p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 74 T^{2} + 7087 T^{4} - 317516 T^{6} + 7087 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 143 T^{2} + 13334 T^{4} - 788712 T^{6} + 13334 p^{2} T^{8} - 143 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 89 T^{2} + 1262 T^{4} + 60840 T^{6} + 1262 p^{2} T^{8} - 89 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 197 T^{2} + 17668 T^{4} + 1129832 T^{6} + 17668 p^{2} T^{8} + 197 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 249 T^{2} + 32150 T^{4} - 2609368 T^{6} + 32150 p^{2} T^{8} - 249 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 201 T^{2} + 16008 T^{4} + 942064 T^{6} + 16008 p^{2} T^{8} + 201 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 174 T^{2} + 17562 T^{4} - 1462506 T^{6} + 17562 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 434 T^{2} + 81242 T^{4} - 8404902 T^{6} + 81242 p^{2} T^{8} - 434 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 14 T + 242 T^{2} - 2128 T^{3} + 242 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 89 | \( ( 1 + 13 T + 248 T^{2} + 2216 T^{3} + 248 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 400 T^{2} + 78800 T^{4} + 9497538 T^{6} + 78800 p^{2} T^{8} + 400 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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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
−3.14636931320669700189782603038, −3.12795869384719935965504048896, −3.08338974784721145729200965376, −2.93580641513700155593137742620, −2.85501251130096518240484267734, −2.71204412475980586370014968621, −2.59523195959021816239205255120, −2.38074057245461762137370995806, −2.30505211552417235856873654667, −2.00225248406143527638167556055, −1.93020131179702585481018945458, −1.85636875041225475953840077410, −1.77786581496343222830654613849, −1.63007954155502941340599682306, −1.32306951815394859920261434686, −1.20505035829378689569198025161, −1.07898266025150840562113641787, −1.04737944331497380141172697531, −0.849844224001897184993589719137, −0.801535055000130591577570168754, −0.58094865506692809708856132822, −0.43860399236413793149704578066, −0.40095514496795487634066397234, −0.24209450148511276822795677935, −0.16285402486445412311851491039,
0.16285402486445412311851491039, 0.24209450148511276822795677935, 0.40095514496795487634066397234, 0.43860399236413793149704578066, 0.58094865506692809708856132822, 0.801535055000130591577570168754, 0.849844224001897184993589719137, 1.04737944331497380141172697531, 1.07898266025150840562113641787, 1.20505035829378689569198025161, 1.32306951815394859920261434686, 1.63007954155502941340599682306, 1.77786581496343222830654613849, 1.85636875041225475953840077410, 1.93020131179702585481018945458, 2.00225248406143527638167556055, 2.30505211552417235856873654667, 2.38074057245461762137370995806, 2.59523195959021816239205255120, 2.71204412475980586370014968621, 2.85501251130096518240484267734, 2.93580641513700155593137742620, 3.08338974784721145729200965376, 3.12795869384719935965504048896, 3.14636931320669700189782603038
Plot not available for L-functions of degree greater than 10.
