| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s − 2·13-s + 16-s − 6·17-s − 19-s + 4·22-s + 6·23-s − 2·26-s + 2·29-s − 6·31-s + 32-s − 6·34-s + 10·37-s − 38-s + 6·43-s + 4·44-s + 6·46-s + 6·47-s − 7·49-s − 2·52-s + 10·53-s + 2·58-s − 2·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.229·19-s + 0.852·22-s + 1.25·23-s − 0.392·26-s + 0.371·29-s − 1.07·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s − 0.162·38-s + 0.914·43-s + 0.603·44-s + 0.884·46-s + 0.875·47-s − 49-s − 0.277·52-s + 1.37·53-s + 0.262·58-s − 0.260·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.466633763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.466633763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.53236935013854916745852823368, −6.95741583040339227289801317529, −6.41406864038512653213894235659, −5.74486049894789382315104243505, −4.80530448554995770659048928417, −4.34477009080680343272114122521, −3.61540647240665124858070471517, −2.67631448874917313784738248942, −1.96161772073748276059947164358, −0.821555384065923307152174126798,
0.821555384065923307152174126798, 1.96161772073748276059947164358, 2.67631448874917313784738248942, 3.61540647240665124858070471517, 4.34477009080680343272114122521, 4.80530448554995770659048928417, 5.74486049894789382315104243505, 6.41406864038512653213894235659, 6.95741583040339227289801317529, 7.53236935013854916745852823368
