| L(s) = 1 | − 4·9-s − 48·31-s − 168·49-s + 144·61-s − 432·79-s − 39·81-s + 624·109-s + 780·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 492·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4/9·9-s − 1.54·31-s − 3.42·49-s + 2.36·61-s − 5.46·79-s − 0.481·81-s + 5.72·109-s + 6.44·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.869817818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869817818\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T^{2} + 55 T^{4} - 56 p^{2} T^{6} + 55 p^{4} T^{8} + 4 p^{8} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 12 p T^{2} + 2535 T^{4} + 76264 T^{6} + 2535 p^{4} T^{8} + 12 p^{9} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 - 390 T^{2} + 91935 T^{4} - 13288020 T^{6} + 91935 p^{4} T^{8} - 390 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 13 | \( ( 1 + 246 T^{2} + 67071 T^{4} + 11718004 T^{6} + 67071 p^{4} T^{8} + 246 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 - 1470 T^{2} + 961599 T^{4} - 358361732 T^{6} + 961599 p^{4} T^{8} - 1470 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 19 | \( ( 1 + 303 T^{2} - 5648 T^{3} + 303 p^{2} T^{4} + p^{6} T^{6} )^{4} \) |
| 23 | \( ( 1 - 2124 T^{2} + 2265255 T^{4} - 1490798808 T^{6} + 2265255 p^{4} T^{8} - 2124 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 630 T^{2} + 49599 T^{4} + 425800332 T^{6} + 49599 p^{4} T^{8} - 630 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 + 12 T + 1191 T^{2} - 8040 T^{3} + 1191 p^{2} T^{4} + 12 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 37 | \( ( 1 + 5622 T^{2} + 15747615 T^{4} + 26937456628 T^{6} + 15747615 p^{4} T^{8} + 5622 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 5730 T^{2} + 17922783 T^{4} - 36601296060 T^{6} + 17922783 p^{4} T^{8} - 5730 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 + 7044 T^{2} + 25527255 T^{4} + 57654006536 T^{6} + 25527255 p^{4} T^{8} + 7044 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 - 7500 T^{2} + 30379143 T^{4} - 80184503000 T^{6} + 30379143 p^{4} T^{8} - 7500 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 - 5454 T^{2} + 24601167 T^{4} - 74943076644 T^{6} + 24601167 p^{4} T^{8} - 5454 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 9030 T^{2} + 39008991 T^{4} - 135788124116 T^{6} + 39008991 p^{4} T^{8} - 9030 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 - 36 T + 10815 T^{2} - 265928 T^{3} + 10815 p^{2} T^{4} - 36 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 67 | \( ( 1 + 21828 T^{2} + 215026743 T^{4} + 1231707761032 T^{6} + 215026743 p^{4} T^{8} + 21828 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 11046 T^{2} + 99426735 T^{4} - 572370011988 T^{6} + 99426735 p^{4} T^{8} - 11046 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 + 5574 T^{2} + 26630895 T^{4} + 31585511956 T^{6} + 26630895 p^{4} T^{8} + 5574 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 + 108 T + 15207 T^{2} + 1113656 T^{3} + 15207 p^{2} T^{4} + 108 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 83 | \( ( 1 - 26748 T^{2} + 362457207 T^{4} - 3108598049016 T^{6} + 362457207 p^{4} T^{8} - 26748 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 19878 T^{2} + 233842479 T^{4} - 1925486462036 T^{6} + 233842479 p^{4} T^{8} - 19878 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 + 43110 T^{2} + 827297295 T^{4} + 9601013122132 T^{6} + 827297295 p^{4} T^{8} + 43110 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
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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
−3.33098062549077850029074550935, −3.14054665788724809332783204402, −3.01558640397469933023045900258, −2.81811134768369245450279132256, −2.75358233087529804059143107502, −2.73216737326831889548166687605, −2.72463525864486073983309632707, −2.52829389847606497412828907473, −2.46460922374427537065572978170, −2.22437573828128517767851800174, −2.09919146711306146099553454641, −2.04636493696708919961635350024, −1.92044814598356739336386145641, −1.74372092870423594006632996297, −1.64888375415577776367758233248, −1.56414329855897695279859393962, −1.33226071022024788810921035178, −1.30525264062961639926406451337, −1.26328109814968183858291089520, −0.926904726034864310051876801979, −0.795180008042637227741125184945, −0.48007488244534129650297865342, −0.39834147643748665494144002692, −0.37191072568896962630656773025, −0.095936438338031723404057212303,
0.095936438338031723404057212303, 0.37191072568896962630656773025, 0.39834147643748665494144002692, 0.48007488244534129650297865342, 0.795180008042637227741125184945, 0.926904726034864310051876801979, 1.26328109814968183858291089520, 1.30525264062961639926406451337, 1.33226071022024788810921035178, 1.56414329855897695279859393962, 1.64888375415577776367758233248, 1.74372092870423594006632996297, 1.92044814598356739336386145641, 2.04636493696708919961635350024, 2.09919146711306146099553454641, 2.22437573828128517767851800174, 2.46460922374427537065572978170, 2.52829389847606497412828907473, 2.72463525864486073983309632707, 2.73216737326831889548166687605, 2.75358233087529804059143107502, 2.81811134768369245450279132256, 3.01558640397469933023045900258, 3.14054665788724809332783204402, 3.33098062549077850029074550935
Plot not available for L-functions of degree greater than 10.
