| L(s) = 1 | + 4-s − 4·11-s − 2·16-s − 8·19-s + 14·29-s + 16·31-s − 26·41-s − 4·44-s + 23·49-s + 4·59-s + 2·61-s + 6·64-s − 20·71-s − 8·76-s − 4·79-s + 18·89-s + 12·101-s − 6·109-s + 14·116-s − 38·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.20·11-s − 1/2·16-s − 1.83·19-s + 2.59·29-s + 2.87·31-s − 4.06·41-s − 0.603·44-s + 23/7·49-s + 0.520·59-s + 0.256·61-s + 3/4·64-s − 2.37·71-s − 0.917·76-s − 0.450·79-s + 1.90·89-s + 1.19·101-s − 0.574·109-s + 1.29·116-s − 3.45·121-s + 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9242285285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9242285285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - T^{2} + 3 T^{4} - 11 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 242 T^{4} - 1811 T^{6} + 242 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 54 T^{2} + 1335 T^{4} - 20836 T^{6} + 1335 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 82 T^{2} + 3087 T^{4} - 67244 T^{6} + 3087 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 63 T^{2} + 2826 T^{4} - 73987 T^{6} + 2826 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 7 T + 2 p T^{2} - 355 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 8 T + 33 T^{2} - 28 T^{3} + 33 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 162 T^{2} + 11847 T^{4} - 534412 T^{6} + 11847 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 13 T + 142 T^{2} + 1069 T^{3} + 142 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 150 T^{2} + 9255 T^{4} - 400228 T^{6} + 9255 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 91 T^{2} + 6942 T^{4} - 321995 T^{6} + 6942 p^{2} T^{8} - 91 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 274 T^{2} + 33303 T^{4} - 2287964 T^{6} + 33303 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 157 T^{2} - 212 T^{3} + 157 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - T + 146 T^{2} - 193 T^{3} + 146 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 203 T^{2} + 26678 T^{4} - 2096795 T^{6} + 26678 p^{2} T^{8} - 203 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 246 T^{2} + 30015 T^{4} - 2521972 T^{6} + 30015 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 2 T + 153 T^{2} + 340 T^{3} + 153 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 327 T^{2} + 49722 T^{4} - 4885459 T^{6} + 49722 p^{2} T^{8} - 327 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 3 T + p T^{2} )^{6} \) |
| 97 | \( 1 - 186 T^{2} + 35727 T^{4} - 3555820 T^{6} + 35727 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.77449953728738061567046157465, −4.62956633447054507569084158917, −4.49226555169357185161401987212, −4.26312346394273320915176980164, −4.18593636334618213743610800257, −4.12385073559852856989567953696, −3.86235977534906597578398889580, −3.72397012784543579599264249356, −3.52616877973154679010957180407, −3.29636667177236693454349687826, −3.01543061017244760675036700718, −2.97022754223013931473739506383, −2.94488472630966496843106278950, −2.52725581465022097263466267836, −2.52634416830408335780319716787, −2.47116709176078679309997969036, −2.13292980739705608243241512653, −2.05413804675348937089212572487, −1.82333040125453357004755081028, −1.53936583071264402875304999262, −1.28343890065068739492833476463, −0.987588809499596905696024685109, −0.923507143846751177706082506293, −0.50382216656992740859692130633, −0.12442749536660047790184468888,
0.12442749536660047790184468888, 0.50382216656992740859692130633, 0.923507143846751177706082506293, 0.987588809499596905696024685109, 1.28343890065068739492833476463, 1.53936583071264402875304999262, 1.82333040125453357004755081028, 2.05413804675348937089212572487, 2.13292980739705608243241512653, 2.47116709176078679309997969036, 2.52634416830408335780319716787, 2.52725581465022097263466267836, 2.94488472630966496843106278950, 2.97022754223013931473739506383, 3.01543061017244760675036700718, 3.29636667177236693454349687826, 3.52616877973154679010957180407, 3.72397012784543579599264249356, 3.86235977534906597578398889580, 4.12385073559852856989567953696, 4.18593636334618213743610800257, 4.26312346394273320915176980164, 4.49226555169357185161401987212, 4.62956633447054507569084158917, 4.77449953728738061567046157465
Plot not available for L-functions of degree greater than 10.
