| L(s) = 1 | − 0.732·2-s − 3-s − 1.46·4-s + 0.732·5-s + 0.732·6-s + 3.73·7-s + 2.53·8-s + 9-s − 0.535·10-s + 4.73·11-s + 1.46·12-s + 13-s − 2.73·14-s − 0.732·15-s + 1.07·16-s + 3.26·17-s − 0.732·18-s + 5.46·19-s − 1.07·20-s − 3.73·21-s − 3.46·22-s − 2.53·24-s − 4.46·25-s − 0.732·26-s − 27-s − 5.46·28-s − 9.66·29-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.577·3-s − 0.732·4-s + 0.327·5-s + 0.298·6-s + 1.41·7-s + 0.896·8-s + 0.333·9-s − 0.169·10-s + 1.42·11-s + 0.422·12-s + 0.277·13-s − 0.730·14-s − 0.189·15-s + 0.267·16-s + 0.792·17-s − 0.172·18-s + 1.25·19-s − 0.239·20-s − 0.814·21-s − 0.738·22-s − 0.517·24-s − 0.892·25-s − 0.143·26-s − 0.192·27-s − 1.03·28-s − 1.79·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321950224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321950224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 9.73T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 8.19T + 89T^{2} \) |
| 97 | \( 1 + 1.07T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.273722835226276255482589970341, −8.888471249866983017775379926225, −7.61985594474241822079543889054, −7.46668596655168341563425077685, −5.88921889767383060529179561493, −5.43750904875789304335230630180, −4.40472903806115616935550056854, −3.73095764450750418493617157698, −1.73504804923049686111992196544, −1.03062855264119002155536407222,
1.03062855264119002155536407222, 1.73504804923049686111992196544, 3.73095764450750418493617157698, 4.40472903806115616935550056854, 5.43750904875789304335230630180, 5.88921889767383060529179561493, 7.46668596655168341563425077685, 7.61985594474241822079543889054, 8.888471249866983017775379926225, 9.273722835226276255482589970341
