| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 22-s + 25-s + 2·26-s + 28-s + 6·29-s − 4·31-s + 32-s + 6·34-s + 35-s + 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.288916287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288916287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.86262975833312581662710655120, −7.19097785542570949823115854286, −6.31265748927539581490144039060, −5.88782902730436016200227905347, −5.12685771260220429872498722488, −4.39065146891315092605231386104, −3.63055737799647593586208005012, −2.83166702719667425071554202582, −1.87985208527121041404981577043, −1.01449688144765976462254563637,
1.01449688144765976462254563637, 1.87985208527121041404981577043, 2.83166702719667425071554202582, 3.63055737799647593586208005012, 4.39065146891315092605231386104, 5.12685771260220429872498722488, 5.88782902730436016200227905347, 6.31265748927539581490144039060, 7.19097785542570949823115854286, 7.86262975833312581662710655120
