| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 11-s + 15-s + 17-s − 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s + 4·31-s − 33-s − 2·35-s + 2·37-s + 6·43-s + 45-s − 3·49-s + 51-s + 6·53-s − 55-s − 14·59-s + 2·61-s − 2·63-s − 14·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.242·17-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s − 0.338·35-s + 0.328·37-s + 0.914·43-s + 0.149·45-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 0.134·55-s − 1.82·59-s + 0.256·61-s − 0.251·63-s − 1.71·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.437263847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437263847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.32017679085073, −12.65749719453333, −12.33729048736249, −11.95605596856742, −11.12254791382922, −10.79077807789568, −10.18142088496252, −9.783341084469215, −9.442951394056983, −8.960547475808588, −8.365892275343526, −7.906914924627569, −7.441800387544292, −6.888258787289763, −6.265755166831558, −5.977240860442821, −5.359748181487782, −4.706368184723167, −4.160800373459549, −3.590538662574585, −3.001490491371568, −2.560756824901060, −1.943509416901192, −1.264792745001273, −0.4294892504538646,
0.4294892504538646, 1.264792745001273, 1.943509416901192, 2.560756824901060, 3.001490491371568, 3.590538662574585, 4.160800373459549, 4.706368184723167, 5.359748181487782, 5.977240860442821, 6.265755166831558, 6.888258787289763, 7.441800387544292, 7.906914924627569, 8.365892275343526, 8.960547475808588, 9.442951394056983, 9.783341084469215, 10.18142088496252, 10.79077807789568, 11.12254791382922, 11.95605596856742, 12.33729048736249, 12.65749719453333, 13.32017679085073
