| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s − 2·13-s + 4·17-s + 4·19-s + 21-s + 6·23-s − 5·25-s + 27-s − 2·29-s + 2·33-s − 6·37-s − 2·39-s + 8·41-s + 8·43-s + 4·47-s + 49-s + 4·51-s − 6·53-s + 4·57-s − 14·61-s + 63-s − 4·67-s + 6·69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.970·17-s + 0.917·19-s + 0.218·21-s + 1.25·23-s − 25-s + 0.192·27-s − 0.371·29-s + 0.348·33-s − 0.986·37-s − 0.320·39-s + 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.79·61-s + 0.125·63-s − 0.488·67-s + 0.722·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994762501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994762501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.41074080313986455036103638000, −9.476665861791711671305936474368, −8.943086252374510058732995262383, −7.71090868218694782736415188362, −7.32082126936645491942637559694, −5.97779432028631552865085959088, −4.97691463353523376663815741347, −3.85252585768054816532884619372, −2.79609515636067281736780650551, −1.36229546664790534828076487479,
1.36229546664790534828076487479, 2.79609515636067281736780650551, 3.85252585768054816532884619372, 4.97691463353523376663815741347, 5.97779432028631552865085959088, 7.32082126936645491942637559694, 7.71090868218694782736415188362, 8.943086252374510058732995262383, 9.476665861791711671305936474368, 10.41074080313986455036103638000
