| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 2·5-s + 6·6-s − 5·7-s − 8·8-s − 18·9-s − 4·10-s + 13·11-s − 12·12-s + 10·14-s − 6·15-s + 16·16-s + 27·17-s + 36·18-s + 75·19-s + 8·20-s + 15·21-s − 26·22-s − 187·23-s + 24·24-s − 121·25-s + 135·27-s − 20·28-s − 13·29-s + 12·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.178·5-s + 0.408·6-s − 0.269·7-s − 0.353·8-s − 2/3·9-s − 0.126·10-s + 0.356·11-s − 0.288·12-s + 0.190·14-s − 0.103·15-s + 1/4·16-s + 0.385·17-s + 0.471·18-s + 0.905·19-s + 0.0894·20-s + 0.155·21-s − 0.251·22-s − 1.69·23-s + 0.204·24-s − 0.967·25-s + 0.962·27-s − 0.134·28-s − 0.0832·29-s + 0.0730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8899690411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8899690411\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 13 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 - 75 T + p^{3} T^{2} \) |
| 23 | \( 1 + 187 T + p^{3} T^{2} \) |
| 29 | \( 1 + 13 T + p^{3} T^{2} \) |
| 31 | \( 1 + 104 T + p^{3} T^{2} \) |
| 37 | \( 1 - 423 T + p^{3} T^{2} \) |
| 41 | \( 1 - 195 T + p^{3} T^{2} \) |
| 43 | \( 1 - 199 T + p^{3} T^{2} \) |
| 47 | \( 1 - 388 T + p^{3} T^{2} \) |
| 53 | \( 1 - 618 T + p^{3} T^{2} \) |
| 59 | \( 1 - 491 T + p^{3} T^{2} \) |
| 61 | \( 1 - 175 T + p^{3} T^{2} \) |
| 67 | \( 1 - 817 T + p^{3} T^{2} \) |
| 71 | \( 1 - 79 T + p^{3} T^{2} \) |
| 73 | \( 1 - 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 764 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1041 T + p^{3} T^{2} \) |
| 97 | \( 1 + p T + p^{3} T^{2} \) |
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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
−11.19136047096466608749981116947, −10.04213578811881431699255150370, −9.438878850230862673888881460217, −8.300357136965507591350731585663, −7.39737177323986201264393299659, −6.14844020743488205589637339806, −5.57643889267372161941465922596, −3.87291395317380628128061631845, −2.37855956086698528037347711579, −0.71853306529523117954593723263,
0.71853306529523117954593723263, 2.37855956086698528037347711579, 3.87291395317380628128061631845, 5.57643889267372161941465922596, 6.14844020743488205589637339806, 7.39737177323986201264393299659, 8.300357136965507591350731585663, 9.438878850230862673888881460217, 10.04213578811881431699255150370, 11.19136047096466608749981116947
