| L(s) = 1 | + 2·2-s − 3-s + 4·4-s + 17·5-s − 2·6-s − 35·7-s + 8·8-s − 26·9-s + 34·10-s + 2·11-s − 4·12-s + 13·13-s − 70·14-s − 17·15-s + 16·16-s − 19·17-s − 52·18-s + 94·19-s + 68·20-s + 35·21-s + 4·22-s − 72·23-s − 8·24-s + 164·25-s + 26·26-s + 53·27-s − 140·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.192·3-s + 1/2·4-s + 1.52·5-s − 0.136·6-s − 1.88·7-s + 0.353·8-s − 0.962·9-s + 1.07·10-s + 0.0548·11-s − 0.0962·12-s + 0.277·13-s − 1.33·14-s − 0.292·15-s + 1/4·16-s − 0.271·17-s − 0.680·18-s + 1.13·19-s + 0.760·20-s + 0.363·21-s + 0.0387·22-s − 0.652·23-s − 0.0680·24-s + 1.31·25-s + 0.196·26-s + 0.377·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.609812354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609812354\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 17 T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 19 T + p^{3} T^{2} \) |
| 19 | \( 1 - 94 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 11 T + p^{3} T^{2} \) |
| 41 | \( 1 + 280 T + p^{3} T^{2} \) |
| 43 | \( 1 - 241 T + p^{3} T^{2} \) |
| 47 | \( 1 - 137 T + p^{3} T^{2} \) |
| 53 | \( 1 + 232 T + p^{3} T^{2} \) |
| 59 | \( 1 + 386 T + p^{3} T^{2} \) |
| 61 | \( 1 - 64 T + p^{3} T^{2} \) |
| 67 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 55 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 - 420 T + p^{3} T^{2} \) |
| 89 | \( 1 + 934 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1154 T + p^{3} T^{2} \) |
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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
−16.80437786070971894766043817922, −15.85286329662653664585933001688, −14.05562219357208349848035310802, −13.40994395178736909586100221211, −12.19007633631656607136146195801, −10.35868729698139241101929943743, −9.261113333992029619217116508410, −6.52009248057915953419005493342, −5.68975293420522781236146597816, −2.95389794381125198681312522075,
2.95389794381125198681312522075, 5.68975293420522781236146597816, 6.52009248057915953419005493342, 9.261113333992029619217116508410, 10.35868729698139241101929943743, 12.19007633631656607136146195801, 13.40994395178736909586100221211, 14.05562219357208349848035310802, 15.85286329662653664585933001688, 16.80437786070971894766043817922
