| L(s) = 1 | + 18·5-s + 8·7-s − 36·11-s − 10·13-s − 18·17-s − 100·19-s − 72·23-s + 199·25-s + 234·29-s − 16·31-s + 144·35-s − 226·37-s − 90·41-s + 452·43-s − 432·47-s − 279·49-s − 414·53-s − 648·55-s + 684·59-s + 422·61-s − 180·65-s + 332·67-s + 360·71-s + 26·73-s − 288·77-s + 512·79-s + 1.18e3·83-s + ⋯ |
| L(s) = 1 | + 1.60·5-s + 0.431·7-s − 0.986·11-s − 0.213·13-s − 0.256·17-s − 1.20·19-s − 0.652·23-s + 1.59·25-s + 1.49·29-s − 0.0926·31-s + 0.695·35-s − 1.00·37-s − 0.342·41-s + 1.60·43-s − 1.34·47-s − 0.813·49-s − 1.07·53-s − 1.58·55-s + 1.50·59-s + 0.885·61-s − 0.343·65-s + 0.605·67-s + 0.601·71-s + 0.0416·73-s − 0.426·77-s + 0.729·79-s + 1.57·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.486297395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486297395\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 - 452 T + p^{3} T^{2} \) |
| 47 | \( 1 + 432 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 422 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1054 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.03770837371975111182857464977, −14.58524432279140748376191649517, −13.62582921617091930672821185849, −12.60221685900743172071963838022, −10.77078057405347183910870874144, −9.826887086999455881132398124603, −8.353103224649821866434014263418, −6.43438705117855584099441326645, −5.06472109844905982513815231224, −2.21712775913447454783926853802,
2.21712775913447454783926853802, 5.06472109844905982513815231224, 6.43438705117855584099441326645, 8.353103224649821866434014263418, 9.826887086999455881132398124603, 10.77078057405347183910870874144, 12.60221685900743172071963838022, 13.62582921617091930672821185849, 14.58524432279140748376191649517, 16.03770837371975111182857464977
