| L(s) = 1 | + 50·5-s − 243·9-s − 664·11-s + 288·19-s + 3.12e3·25-s − 1.89e3·29-s − 1.02e4·31-s − 1.32e3·41-s − 1.21e4·45-s + 6.46e4·49-s − 3.32e4·55-s − 1.02e4·59-s − 5.21e4·61-s − 2.82e4·71-s + 2.20e5·79-s + 3.93e4·81-s + 3.51e5·89-s + 1.44e4·95-s + 1.61e5·99-s − 4.96e5·101-s + 8.47e5·109-s − 4.86e5·121-s − 9.60e4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 1.65·11-s + 0.183·19-s + 25-s − 0.417·29-s − 1.91·31-s − 0.123·41-s − 0.894·45-s + 3.84·49-s − 1.47·55-s − 0.385·59-s − 1.79·61-s − 0.665·71-s + 3.96·79-s + 2/3·81-s + 4.70·89-s + 0.163·95-s + 1.65·99-s − 4.84·101-s + 6.82·109-s − 3.01·121-s − 0.549·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.048162638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.048162638\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{4} T^{2} )^{3} \) |
| 5 | \( 1 - 2 p^{2} T - p^{4} T^{2} + 2268 p^{3} T^{3} - p^{9} T^{4} - 2 p^{12} T^{5} + p^{15} T^{6} \) |
| good | 7 | \( 1 - 64626 T^{2} + 2028003087 T^{4} - 41058323262556 T^{6} + 2028003087 p^{10} T^{8} - 64626 p^{20} T^{10} + p^{30} T^{12} \) |
| 11 | \( ( 1 + 332 T + 408509 T^{2} + 107689464 T^{3} + 408509 p^{5} T^{4} + 332 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 13 | \( 1 - 2165574 T^{2} + 1976112655671 T^{4} - 972719104225889812 T^{6} + 1976112655671 p^{10} T^{8} - 2165574 p^{20} T^{10} + p^{30} T^{12} \) |
| 17 | \( 1 - 6393390 T^{2} + 19617942041247 T^{4} - 35338776492878157220 T^{6} + 19617942041247 p^{10} T^{8} - 6393390 p^{20} T^{10} + p^{30} T^{12} \) |
| 19 | \( ( 1 - 144 T + 1911561 T^{2} + 1871015072 T^{3} + 1911561 p^{5} T^{4} - 144 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 23 | \( 1 - 558582 p T^{2} + 119828345726079 T^{4} - \)\(79\!\cdots\!56\)\( T^{6} + 119828345726079 p^{10} T^{8} - 558582 p^{21} T^{10} + p^{30} T^{12} \) |
| 29 | \( ( 1 + 946 T + 36417431 T^{2} + 76110228372 T^{3} + 36417431 p^{5} T^{4} + 946 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 31 | \( ( 1 + 5124 T + 61295997 T^{2} + 221905216248 T^{3} + 61295997 p^{5} T^{4} + 5124 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 37 | \( 1 - 290752854 T^{2} + 38769784471186119 T^{4} - \)\(32\!\cdots\!24\)\( T^{6} + 38769784471186119 p^{10} T^{8} - 290752854 p^{20} T^{10} + p^{30} T^{12} \) |
| 41 | \( ( 1 + 662 T + 143511479 T^{2} + 442940770164 T^{3} + 143511479 p^{5} T^{4} + 662 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 43 | \( 1 - 502572210 T^{2} + 127464803683947447 T^{4} - \)\(22\!\cdots\!80\)\( T^{6} + 127464803683947447 p^{10} T^{8} - 502572210 p^{20} T^{10} + p^{30} T^{12} \) |
| 47 | \( 1 - 288722058 T^{2} + 116437215277524783 T^{4} - \)\(17\!\cdots\!64\)\( T^{6} + 116437215277524783 p^{10} T^{8} - 288722058 p^{20} T^{10} + p^{30} T^{12} \) |
| 53 | \( 1 - 1963456710 T^{2} + 1786491383741543655 T^{4} - \)\(95\!\cdots\!64\)\( T^{6} + 1786491383741543655 p^{10} T^{8} - 1963456710 p^{20} T^{10} + p^{30} T^{12} \) |
| 59 | \( ( 1 + 5148 T + 597505773 T^{2} + 370643378904 T^{3} + 597505773 p^{5} T^{4} + 5148 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 61 | \( ( 1 + 26058 T + 886088883 T^{2} + 13089282520252 T^{3} + 886088883 p^{5} T^{4} + 26058 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 67 | \( 1 - 7507145634 T^{2} + 24155236685124504999 T^{4} - \)\(42\!\cdots\!84\)\( T^{6} + 24155236685124504999 p^{10} T^{8} - 7507145634 p^{20} T^{10} + p^{30} T^{12} \) |
| 71 | \( ( 1 + 14144 T + 2246377637 T^{2} + 195098790528 p T^{3} + 2246377637 p^{5} T^{4} + 14144 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 73 | \( 1 - 2138297622 T^{2} + 9645229369668502143 T^{4} - \)\(18\!\cdots\!56\)\( T^{6} + 9645229369668502143 p^{10} T^{8} - 2138297622 p^{20} T^{10} + p^{30} T^{12} \) |
| 79 | \( ( 1 - 110100 T + 12028730445 T^{2} - 673264081259096 T^{3} + 12028730445 p^{5} T^{4} - 110100 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 83 | \( 1 - 5878219362 T^{2} + 30391064056477835847 T^{4} - \)\(17\!\cdots\!76\)\( T^{6} + 30391064056477835847 p^{10} T^{8} - 5878219362 p^{20} T^{10} + p^{30} T^{12} \) |
| 89 | \( ( 1 - 175710 T + 24190100439 T^{2} - 1951044010529988 T^{3} + 24190100439 p^{5} T^{4} - 175710 p^{10} T^{5} + p^{15} T^{6} )^{2} \) |
| 97 | \( 1 - 18122177670 T^{2} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + \)\(22\!\cdots\!47\)\( p^{10} T^{8} - 18122177670 p^{20} T^{10} + p^{30} T^{12} \) |
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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
−6.42582129617523744449456580981, −6.25766003568171610653178322454, −6.08526685673050354820450443270, −6.00356548035327835097324882156, −5.43417153306926416711772439643, −5.36237023642681970124116237716, −5.21893632173749865458845505284, −5.20734963208991396037238028837, −5.14358375231149841087908823931, −4.35024463092622852528000318586, −4.31942420039526035046985094876, −4.12712658511643339861966407715, −3.87576034416872768577615001423, −3.31562636061074473763893553931, −3.03366645458857037602418252665, −3.02815565891587064538755136238, −2.99732660512568948067387872655, −2.23976773708647889466540825462, −2.06896460319291925866186484653, −1.95349599294604227904756624032, −1.89160739532275905641488405071, −1.09123725135891072925621466639, −0.69186951617261069637196147054, −0.60743233267377570796642129166, −0.29872826552891485273566752321,
0.29872826552891485273566752321, 0.60743233267377570796642129166, 0.69186951617261069637196147054, 1.09123725135891072925621466639, 1.89160739532275905641488405071, 1.95349599294604227904756624032, 2.06896460319291925866186484653, 2.23976773708647889466540825462, 2.99732660512568948067387872655, 3.02815565891587064538755136238, 3.03366645458857037602418252665, 3.31562636061074473763893553931, 3.87576034416872768577615001423, 4.12712658511643339861966407715, 4.31942420039526035046985094876, 4.35024463092622852528000318586, 5.14358375231149841087908823931, 5.20734963208991396037238028837, 5.21893632173749865458845505284, 5.36237023642681970124116237716, 5.43417153306926416711772439643, 6.00356548035327835097324882156, 6.08526685673050354820450443270, 6.25766003568171610653178322454, 6.42582129617523744449456580981
Plot not available for L-functions of degree greater than 10.
