| L(s) = 1 | + 4·9-s + 18·25-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 + ⋯ |
| L(s) = 1 | + 4/9·9-s + 0.719·25-s − 1.54·31-s + 24/7·49-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(19.13578646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.13578646\) |
| \(L(2)\) |
|
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 | \( 1 - 18 T^{2} - 369 T^{4} + 36 p^{4} T^{6} - 369 p^{4} T^{8} - 18 p^{8} T^{10} + p^{12} T^{12} \) |
| 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
−3.06042193919946369042968085488, −2.83782001641511801366556214754, −2.75280266846992939386652767783, −2.71695839540930929848231415483, −2.65459373600714464225044558581, −2.45475921884043351937876890429, −2.16234170622973902475085151534, −2.14377796849477621941997307018, −2.12657613726709766239178069063, −2.12404842130682739947658665628, −1.99679206519668328220624525649, −1.88489871056942991535079685035, −1.82858198534042788965404352427, −1.72567565732727203672750615814, −1.48621821255930266497850030368, −1.20707123119562295836735864718, −1.10966194419499079426373158506, −1.07502980981406236658271039313, −1.05776588815820831278498948416, −0.862406611255538759992944275436, −0.797755670326124786965834965045, −0.49801111762467393571392888499, −0.34229547672049848689474564291, −0.31634017418413208575156666394, −0.22118345853826438440072569220,
0.22118345853826438440072569220, 0.31634017418413208575156666394, 0.34229547672049848689474564291, 0.49801111762467393571392888499, 0.797755670326124786965834965045, 0.862406611255538759992944275436, 1.05776588815820831278498948416, 1.07502980981406236658271039313, 1.10966194419499079426373158506, 1.20707123119562295836735864718, 1.48621821255930266497850030368, 1.72567565732727203672750615814, 1.82858198534042788965404352427, 1.88489871056942991535079685035, 1.99679206519668328220624525649, 2.12404842130682739947658665628, 2.12657613726709766239178069063, 2.14377796849477621941997307018, 2.16234170622973902475085151534, 2.45475921884043351937876890429, 2.65459373600714464225044558581, 2.71695839540930929848231415483, 2.75280266846992939386652767783, 2.83782001641511801366556214754, 3.06042193919946369042968085488
Plot not available for L-functions of degree greater than 10.
