| L(s) = 1 | + 1.02e3·2-s + 1.04e6·4-s + 1.93e7·5-s + 1.07e9·8-s + 1.97e10·10-s + 1.90e11·13-s + 1.09e12·16-s − 7.50e11·17-s + 2.02e13·20-s + 2.77e14·25-s + 1.95e14·26-s − 2.03e14·29-s + 1.12e15·32-s − 7.68e14·34-s − 9.49e15·37-s + 2.07e16·40-s − 1.60e16·41-s + 7.97e16·49-s + 2.84e17·50-s + 2.00e17·52-s − 2.63e17·53-s − 2.08e17·58-s + 3.42e17·61-s + 1.15e18·64-s + 3.68e18·65-s − 7.86e17·68-s + 5.39e18·73-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.97·5-s + 8-s + 1.97·10-s + 1.38·13-s + 16-s − 0.372·17-s + 1.97·20-s + 2.90·25-s + 1.38·26-s − 0.482·29-s + 32-s − 0.372·34-s − 1.97·37-s + 1.97·40-s − 1.19·41-s + 49-s + 2.90·50-s + 1.38·52-s − 1.50·53-s − 0.482·58-s + 0.480·61-s + 64-s + 2.73·65-s − 0.372·68-s + 1.25·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(7.363491106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.363491106\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{10} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 19306574 T + p^{20} T^{2} \) |
| 7 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 11 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 13 | \( 1 - 190840318802 T + p^{20} T^{2} \) |
| 17 | \( 1 + 750325121602 T + p^{20} T^{2} \) |
| 19 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 23 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 29 | \( 1 + 203154876160402 T + p^{20} T^{2} \) |
| 31 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 37 | \( 1 + 9492206529013198 T + p^{20} T^{2} \) |
| 41 | \( 1 + 16082418088944802 T + p^{20} T^{2} \) |
| 43 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 47 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 53 | \( 1 + 263609364120076402 T + p^{20} T^{2} \) |
| 59 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 61 | \( 1 - 342453856112605202 T + p^{20} T^{2} \) |
| 67 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 71 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 73 | \( 1 - 5395059597962887202 T + p^{20} T^{2} \) |
| 79 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 83 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 89 | \( 1 + 10944684939688527202 T + p^{20} T^{2} \) |
| 97 | \( 1 + 72063723240789129598 T + p^{20} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.64885043965407402381280781971, −11.04630242232801766762372466496, −10.11854551139174254093166006881, −8.756631751250490375291448587050, −6.77754871485847122503493808596, −5.95686846109133091642167305718, −5.05599293564767978900800390806, −3.42722295848985864329315600029, −2.12501251193559886303581448392, −1.34065076913834599704352076145,
1.34065076913834599704352076145, 2.12501251193559886303581448392, 3.42722295848985864329315600029, 5.05599293564767978900800390806, 5.95686846109133091642167305718, 6.77754871485847122503493808596, 8.756631751250490375291448587050, 10.11854551139174254093166006881, 11.04630242232801766762372466496, 12.64885043965407402381280781971
