| L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s + 8-s + 6·9-s + 10-s − 2·11-s + 3·12-s + 3·15-s + 16-s + 3·17-s + 6·18-s + 8·19-s + 20-s − 2·22-s − 4·23-s + 3·24-s − 4·25-s + 9·27-s − 5·29-s + 3·30-s + 3·31-s + 32-s − 6·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s + 0.866·12-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.83·19-s + 0.223·20-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 4/5·25-s + 1.73·27-s − 0.928·29-s + 0.547·30-s + 0.538·31-s + 0.176·32-s − 1.04·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.461854347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.461854347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.123557951705472949061335124821, −7.85026172638606187205534601340, −7.23231108072034238694742458172, −6.14524423168371222824775946693, −5.40751460278228644927408637581, −4.51028341699283754401481648870, −3.57808110978732908807890043221, −3.06234607139059152508576150169, −2.26690233317394818814059857958, −1.40185633314711735542351126186,
1.40185633314711735542351126186, 2.26690233317394818814059857958, 3.06234607139059152508576150169, 3.57808110978732908807890043221, 4.51028341699283754401481648870, 5.40751460278228644927408637581, 6.14524423168371222824775946693, 7.23231108072034238694742458172, 7.85026172638606187205534601340, 8.123557951705472949061335124821
