| L(s) = 1 | − 4·3-s + 4-s + 2·9-s − 4·12-s − 8·13-s − 2·16-s + 8·17-s + 16·23-s − 3·25-s + 8·27-s − 4·29-s + 2·36-s + 32·39-s + 24·43-s + 8·48-s + 22·49-s − 32·51-s − 8·52-s + 12·53-s − 12·61-s + 6·64-s + 8·68-s − 64·69-s + 12·75-s + 24·79-s + 5·81-s + 16·87-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2/3·9-s − 1.15·12-s − 2.21·13-s − 1/2·16-s + 1.94·17-s + 3.33·23-s − 3/5·25-s + 1.53·27-s − 0.742·29-s + 1/3·36-s + 5.12·39-s + 3.65·43-s + 1.15·48-s + 22/7·49-s − 4.48·51-s − 1.10·52-s + 1.64·53-s − 1.53·61-s + 3/4·64-s + 0.970·68-s − 7.70·69-s + 1.38·75-s + 2.70·79-s + 5/9·81-s + 1.71·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2304581073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2304581073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 + 8 T + 3 p T^{2} + 160 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - T^{2} + 3 T^{4} - 11 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 + 2 T + 5 T^{2} + 10 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 22 T^{2} + 41 p T^{4} - 2404 T^{6} + 41 p^{3} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 18 T^{2} + 387 T^{4} - 3904 T^{6} + 387 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 4 T + 19 T^{2} - 40 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 70 T^{2} + 2267 T^{4} - 49288 T^{6} + 2267 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 8 T + 85 T^{2} - 374 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 79 T^{2} + 104 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 130 T^{2} + 8075 T^{4} - 310336 T^{6} + 8075 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 70 T^{2} + 4295 T^{4} - 175492 T^{6} + 4295 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 154 T^{2} + 12351 T^{4} - 627692 T^{6} + 12351 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 12 T + 141 T^{2} - 1034 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 262 T^{2} + 29487 T^{4} - 1821764 T^{6} + 29487 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 6 T + 99 T^{2} - 708 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 270 T^{2} + 32859 T^{4} - 2408056 T^{6} + 32859 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 171 T^{2} + 656 T^{3} + 171 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 246 T^{2} + 28695 T^{4} - 2226404 T^{6} + 28695 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 382 T^{2} + 63315 T^{4} - 5855192 T^{6} + 63315 p^{2} T^{8} - 382 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 142 T^{2} + 16847 T^{4} - 1458004 T^{6} + 16847 p^{2} T^{8} - 142 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 189 T^{2} - 1864 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 342 T^{2} + 57303 T^{4} - 5942500 T^{6} + 57303 p^{2} T^{8} - 342 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 342 T^{2} + 59679 T^{4} - 6531892 T^{6} + 59679 p^{2} T^{8} - 342 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 442 T^{2} + 88271 T^{4} - 10631788 T^{6} + 88271 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−8.850415962085183974280220793406, −8.099435753844659029482053967235, −8.035550335690015358908252363190, −7.79155638619228010991487869864, −7.75043303720510673533145671322, −7.27787089118481219684821541918, −7.26603661263541042190303201899, −6.86196605652434717143045465980, −6.73234369574511293236824621933, −6.65910692770411291565565913649, −6.14585082942880469777021637917, −5.79122070531119725550367002383, −5.51102560512337133794403396456, −5.46288159275398275192089909136, −5.44998258684393341157446975988, −5.37098552289758347766473307767, −4.86321361822595557785108330774, −4.48298773849430596679237318785, −4.04651105794086478505517210749, −3.96352922720447097511227539398, −3.32437052564799633022118231743, −2.80712830311512006685651363026, −2.48887610981466579988293126407, −2.46972928728870805519330112331, −1.01415196881671313783806455603,
1.01415196881671313783806455603, 2.46972928728870805519330112331, 2.48887610981466579988293126407, 2.80712830311512006685651363026, 3.32437052564799633022118231743, 3.96352922720447097511227539398, 4.04651105794086478505517210749, 4.48298773849430596679237318785, 4.86321361822595557785108330774, 5.37098552289758347766473307767, 5.44998258684393341157446975988, 5.46288159275398275192089909136, 5.51102560512337133794403396456, 5.79122070531119725550367002383, 6.14585082942880469777021637917, 6.65910692770411291565565913649, 6.73234369574511293236824621933, 6.86196605652434717143045465980, 7.26603661263541042190303201899, 7.27787089118481219684821541918, 7.75043303720510673533145671322, 7.79155638619228010991487869864, 8.035550335690015358908252363190, 8.099435753844659029482053967235, 8.850415962085183974280220793406
Plot not available for L-functions of degree greater than 10.
