L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 2·13-s − 14-s + 16-s + 2·17-s + 4·19-s + 20-s − 22-s + 4·23-s + 25-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s + 2·34-s − 35-s − 2·37-s + 4·38-s + 40-s − 10·41-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 + 0.485·17-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-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 + 0.342·34-s − 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s − 1.56·41-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
\(L(1)\) |
\(\approx\) |
\(3.471652639\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.471652639\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.72733511771100122956288396712, −7.13787326545908489449772211861, −6.54934924541732159697259654405, −5.63742114106130401870868544336, −5.23495886550686119665212148094, −4.47413291142266428474632299932, −3.44892053222614935528814277043, −2.89574165775395774445459182491, −2.01224527567041426439840040956, −0.860794665025687162150817382833,
0.860794665025687162150817382833, 2.01224527567041426439840040956, 2.89574165775395774445459182491, 3.44892053222614935528814277043, 4.47413291142266428474632299932, 5.23495886550686119665212148094, 5.63742114106130401870868544336, 6.54934924541732159697259654405, 7.13787326545908489449772211861, 7.72733511771100122956288396712