| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s + 6·11-s − 12-s − 13-s + 14-s + 16-s + 3·17-s − 2·18-s + 2·19-s − 21-s + 6·22-s − 24-s − 26-s + 5·27-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.80·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.458·19-s − 0.218·21-s + 1.27·22-s − 0.204·24-s − 0.196·26-s + 0.962·27-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.04·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075155991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075155991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.85087011487847571021356695136, −9.770541951432861923539732431066, −8.847558044717209987907575551843, −7.77590002125487879128863474115, −6.69237709441841342358567965525, −6.00755235942212119970833220848, −5.08129376034999934547621265049, −4.10681052814243994594072799462, −2.97165737705228818597115826999, −1.31126410158638310414896081400,
1.31126410158638310414896081400, 2.97165737705228818597115826999, 4.10681052814243994594072799462, 5.08129376034999934547621265049, 6.00755235942212119970833220848, 6.69237709441841342358567965525, 7.77590002125487879128863474115, 8.847558044717209987907575551843, 9.770541951432861923539732431066, 10.85087011487847571021356695136
