| L(s) = 1 | + 4-s + 76·11-s + 4·16-s − 374·19-s − 320·29-s + 454·31-s − 676·41-s + 76·44-s + 1.22e3·49-s − 280·59-s + 1.19e3·61-s + 819·64-s − 1.20e3·71-s − 374·76-s − 1.25e3·79-s − 4.30e3·89-s − 1.36e3·101-s − 5.21e3·109-s − 320·116-s + 1.57e3·121-s + 454·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/8·4-s + 2.08·11-s + 1/16·16-s − 4.51·19-s − 2.04·29-s + 2.63·31-s − 2.57·41-s + 0.260·44-s + 3.57·49-s − 0.617·59-s + 2.49·61-s + 1.59·64-s − 2.01·71-s − 0.564·76-s − 1.79·79-s − 5.13·89-s − 1.34·101-s − 4.58·109-s − 0.256·116-s + 1.17·121-s + 0.328·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.367657454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367657454\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - T^{2} - 3 T^{4} - 203 p^{2} T^{6} - 3 p^{6} T^{8} - p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 1226 T^{2} + 769535 T^{4} - 316892876 T^{6} + 769535 p^{6} T^{8} - 1226 p^{12} T^{10} + p^{18} T^{12} \) |
| 11 | \( ( 1 - 38 T + 1381 T^{2} - 17876 T^{3} + 1381 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 13 | \( 1 - 3246 T^{2} + 11631063 T^{4} - 30956231140 T^{6} + 11631063 p^{6} T^{8} - 3246 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 6163 T^{2} + 56646390 T^{4} - 189472696871 T^{6} + 56646390 p^{6} T^{8} - 6163 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 + 187 T + 24164 T^{2} + 2039395 T^{3} + 24164 p^{3} T^{4} + 187 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 19839 T^{2} + 507262974 T^{4} - 5894416200139 T^{6} + 507262974 p^{6} T^{8} - 19839 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 160 T + 25399 T^{2} - 88280 T^{3} + 25399 p^{3} T^{4} + 160 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 - 227 T + 71400 T^{2} - 13771435 T^{3} + 71400 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 97986 T^{2} + 8873792007 T^{4} - 470111365900348 T^{6} + 8873792007 p^{6} T^{8} - 97986 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 338 T + 163951 T^{2} + 34473956 T^{3} + 163951 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 156726 T^{2} + 19336556583 T^{4} - 1757584565934100 T^{6} + 19336556583 p^{6} T^{8} - 156726 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 508570 T^{2} + 117397962543 T^{4} - 15651426994430252 T^{6} + 117397962543 p^{6} T^{8} - 508570 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 744595 T^{2} + 248680653006 T^{4} - 47637358185692759 T^{6} + 248680653006 p^{6} T^{8} - 744595 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 + 140 T + 449689 T^{2} + 23374640 T^{3} + 449689 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 1185590 T^{2} + 637803099863 T^{4} - 223447532584291412 T^{6} + 637803099863 p^{6} T^{8} - 1185590 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 602 T + 490081 T^{2} + 150373964 T^{3} + 490081 p^{3} T^{4} + 602 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 467850 T^{2} + 420984954303 T^{4} - 140390583242060428 T^{6} + 420984954303 p^{6} T^{8} - 467850 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 + 629 T + 1576176 T^{2} + 622253365 T^{3} + 1576176 p^{3} T^{4} + 629 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 1042599 T^{2} + 467625790374 T^{4} - 106913777507688499 T^{6} + 467625790374 p^{6} T^{8} - 1042599 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 2154 T + 3172479 T^{2} + 3111332052 T^{3} + 3172479 p^{3} T^{4} + 2154 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 4626246 T^{2} + 9511067662287 T^{4} - 11170340228400453268 T^{6} + 9511067662287 p^{6} T^{8} - 4626246 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.23199903959049901851841153383, −5.15606432488061929765034508022, −4.54741818266149845123578345098, −4.50970882978343909782239815263, −4.39741285259226458252218332700, −4.30179940037128473431029182355, −4.15638810995570497326385646030, −4.00476771587636696514289442048, −3.79709724374221491828826995917, −3.68637661612565338784043519880, −3.64927030376496375798406338730, −3.02026810687889175972061363407, −2.88112545707199655555854892498, −2.77327484339063949094144368395, −2.66642465873837629777824048395, −2.40269303659349924327912697970, −1.95236711675211216672998741928, −1.87323775722988579917648050132, −1.76175452185414406853943400895, −1.67706990240062772774698288754, −1.09589175438936296242091522294, −1.08852767376218384787376887657, −0.73932677432079617476215513529, −0.29955255620455492871205772705, −0.14747141585450577505130855490,
0.14747141585450577505130855490, 0.29955255620455492871205772705, 0.73932677432079617476215513529, 1.08852767376218384787376887657, 1.09589175438936296242091522294, 1.67706990240062772774698288754, 1.76175452185414406853943400895, 1.87323775722988579917648050132, 1.95236711675211216672998741928, 2.40269303659349924327912697970, 2.66642465873837629777824048395, 2.77327484339063949094144368395, 2.88112545707199655555854892498, 3.02026810687889175972061363407, 3.64927030376496375798406338730, 3.68637661612565338784043519880, 3.79709724374221491828826995917, 4.00476771587636696514289442048, 4.15638810995570497326385646030, 4.30179940037128473431029182355, 4.39741285259226458252218332700, 4.50970882978343909782239815263, 4.54741818266149845123578345098, 5.15606432488061929765034508022, 5.23199903959049901851841153383
Plot not available for L-functions of degree greater than 10.
