| L(s) = 1 | + 48·17-s + 114·25-s + 336·41-s + 318·49-s + 60·73-s + 1.24e3·89-s − 204·97-s + 1.20e3·113-s − 618·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 156·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.82·17-s + 4.55·25-s + 8.19·41-s + 6.48·49-s + 0.821·73-s + 14.0·89-s − 2.10·97-s + 10.6·113-s − 5.10·121-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 + 0.923·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(272.7666425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(272.7666425\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 57 T^{2} + 1794 T^{4} - 41897 T^{6} + 1794 p^{4} T^{8} - 57 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 7 | \( ( 1 - 159 T^{2} + 14466 T^{4} - 864799 T^{6} + 14466 p^{4} T^{8} - 159 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 + 309 T^{2} + 40998 T^{4} + 4295297 T^{6} + 40998 p^{4} T^{8} + 309 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 13 | \( ( 1 - 6 p T^{2} + 87 p^{2} T^{4} - 9725348 T^{6} + 87 p^{6} T^{8} - 6 p^{9} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 - 12 T + 327 T^{2} - 24 T^{3} + 327 p^{2} T^{4} - 12 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 19 | \( ( 1 + 510 T^{2} + 235311 T^{4} + 79937732 T^{6} + 235311 p^{4} T^{8} + 510 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 23 | \( ( 1 - 654 T^{2} + 784671 T^{4} - 690596 p^{2} T^{6} + 784671 p^{4} T^{8} - 654 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 3534 T^{2} + 5907759 T^{4} - 6107279780 T^{6} + 5907759 p^{4} T^{8} - 3534 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 - 2319 T^{2} + 3460386 T^{4} - 3880559887 T^{6} + 3460386 p^{4} T^{8} - 2319 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2958 T^{2} + 6236943 T^{4} - 8446524068 T^{6} + 6236943 p^{4} T^{8} - 2958 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 84 T + 4071 T^{2} - 169224 T^{3} + 4071 p^{2} T^{4} - 84 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 43 | \( ( 1 + 6774 T^{2} + 22503903 T^{4} + 49221757172 T^{6} + 22503903 p^{4} T^{8} + 6774 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 - 5766 T^{2} + 22375887 T^{4} - 58361753108 T^{6} + 22375887 p^{4} T^{8} - 5766 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 - 15753 T^{2} + 106280994 T^{4} - 392809110617 T^{6} + 106280994 p^{4} T^{8} - 15753 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 + 9654 T^{2} + 47947551 T^{4} + 179050130804 T^{6} + 47947551 p^{4} T^{8} + 9654 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{6} \) |
| 67 | \( ( 1 + 20598 T^{2} + 198533823 T^{4} + 1132271795828 T^{6} + 198533823 p^{4} T^{8} + 20598 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 7854 T^{2} + 6574047 T^{4} + 80774387548 T^{6} + 6574047 p^{4} T^{8} - 7854 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 - 15 T + 10578 T^{2} - 280591 T^{3} + 10578 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 79 | \( ( 1 - 28578 T^{2} + 376929087 T^{4} - 474751804 p^{2} T^{6} + 376929087 p^{4} T^{8} - 28578 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 83 | \( ( 1 + 10533 T^{2} - 37222170 T^{4} - 869901312751 T^{6} - 37222170 p^{4} T^{8} + 10533 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 312 T + 52707 T^{2} - 5623536 T^{3} + 52707 p^{2} T^{4} - 312 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 97 | \( ( 1 + 51 T + 5862 T^{2} - 251561 T^{3} + 5862 p^{2} T^{4} + 51 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
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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
−2.54119075221654542129724620264, −2.49489127102071754915611727844, −2.38953317849758998056217894360, −2.37012497923229409354394392700, −2.29698228045456751289608927506, −2.21967866117901880485918390575, −2.17307358079332710620423906057, −2.12961066210105318447015868143, −2.06289048567130608104652228958, −1.92298122522672124003944816769, −1.76511987725358158763964369159, −1.74072520443816210325187895143, −1.42209983941032298892833949482, −1.20066916941258965829655199876, −1.12791845915317175462117261669, −1.04776613325914870916574091514, −0.899319676586355240772529861537, −0.898849009599420351185481158453, −0.822609075256474273486912736573, −0.78725293874372804481801919003, −0.74298359102707909776793503602, −0.67829378021738169268568044390, −0.65953633221048804331524976978, −0.29201619215806631023600027291, −0.25955187183058172600715307462,
0.25955187183058172600715307462, 0.29201619215806631023600027291, 0.65953633221048804331524976978, 0.67829378021738169268568044390, 0.74298359102707909776793503602, 0.78725293874372804481801919003, 0.822609075256474273486912736573, 0.898849009599420351185481158453, 0.899319676586355240772529861537, 1.04776613325914870916574091514, 1.12791845915317175462117261669, 1.20066916941258965829655199876, 1.42209983941032298892833949482, 1.74072520443816210325187895143, 1.76511987725358158763964369159, 1.92298122522672124003944816769, 2.06289048567130608104652228958, 2.12961066210105318447015868143, 2.17307358079332710620423906057, 2.21967866117901880485918390575, 2.29698228045456751289608927506, 2.37012497923229409354394392700, 2.38953317849758998056217894360, 2.49489127102071754915611727844, 2.54119075221654542129724620264
Plot not available for L-functions of degree greater than 10.
