| L(s) = 1 | − 2·5-s + 6·9-s + 4·11-s − 4·19-s + 25-s + 12·29-s − 16·31-s + 4·41-s − 12·45-s + 18·49-s − 8·55-s + 4·59-s − 20·61-s + 24·71-s + 32·79-s + 13·81-s + 20·89-s + 8·95-s + 24·99-s + 12·101-s − 52·109-s − 14·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2·9-s + 1.20·11-s − 0.917·19-s + 1/5·25-s + 2.22·29-s − 2.87·31-s + 0.624·41-s − 1.78·45-s + 18/7·49-s − 1.07·55-s + 0.520·59-s − 2.56·61-s + 2.84·71-s + 3.60·79-s + 13/9·81-s + 2.11·89-s + 0.820·95-s + 2.41·99-s + 1.19·101-s − 4.98·109-s − 1.27·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.047527734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.047527734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - 2 p T^{2} + 23 T^{4} - 68 T^{6} + 23 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 18 T^{2} + 239 T^{4} - 1868 T^{6} + 239 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 30 T^{2} + 359 T^{4} - 3332 T^{6} + 359 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 54 T^{2} + 1583 T^{4} - 31412 T^{6} + 1583 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 21 T^{2} - 28 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 98 T^{2} + 4335 T^{4} - 120044 T^{6} + 4335 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 94 T^{2} + 6967 T^{4} - 285508 T^{6} + 6967 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 102 T^{2} + 4103 T^{4} - 124676 T^{6} + 4103 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 210 T^{2} + 21311 T^{4} - 1269644 T^{6} + 21311 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 206 T^{2} + 21015 T^{4} - 1361636 T^{6} + 21015 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 141 T^{2} - 132 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 10 T + 155 T^{2} + 1212 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 182 T^{2} + 24055 T^{4} - 1826084 T^{6} + 24055 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 12 T + 197 T^{2} - 1384 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 38 T^{2} + 10495 T^{4} - 503444 T^{6} + 10495 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 16 T + 269 T^{2} - 2400 T^{3} + 269 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 118 T^{2} + 13335 T^{4} - 1456804 T^{6} + 13335 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 10 T + 151 T^{2} - 684 T^{3} + 151 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 278 T^{2} + 46735 T^{4} - 5290868 T^{6} + 46735 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.67091714590042745504455269431, −5.37676412760819235946041196478, −5.31651097508777487429847826360, −5.09049726672864314994267789092, −4.85867903245936313465846955545, −4.76958914249168510742625634764, −4.67942895187311250777996678728, −4.46370220917528219894505116518, −4.07984441611559768358598233867, −4.00376065829358913741521227302, −3.86475817639767833167489728447, −3.80636356463802694250025123771, −3.64050511677064423155726932241, −3.59224656910736929562405462684, −3.05475489488875960938806754662, −3.01728566509559004234675024774, −2.45357920631419073621964914111, −2.38264388151561916408534210290, −2.34784486574844601187947777581, −2.03547597235539946391631485185, −1.49636986216914167016105981809, −1.43111198691620754670780777606, −1.21745518006587487699192510333, −0.837512602725564302670678358522, −0.37157634925857915488233245041,
0.37157634925857915488233245041, 0.837512602725564302670678358522, 1.21745518006587487699192510333, 1.43111198691620754670780777606, 1.49636986216914167016105981809, 2.03547597235539946391631485185, 2.34784486574844601187947777581, 2.38264388151561916408534210290, 2.45357920631419073621964914111, 3.01728566509559004234675024774, 3.05475489488875960938806754662, 3.59224656910736929562405462684, 3.64050511677064423155726932241, 3.80636356463802694250025123771, 3.86475817639767833167489728447, 4.00376065829358913741521227302, 4.07984441611559768358598233867, 4.46370220917528219894505116518, 4.67942895187311250777996678728, 4.76958914249168510742625634764, 4.85867903245936313465846955545, 5.09049726672864314994267789092, 5.31651097508777487429847826360, 5.37676412760819235946041196478, 5.67091714590042745504455269431
Plot not available for L-functions of degree greater than 10.
