L(s) = 1 | + 3-s − 5-s − 2·9-s − 3·11-s + 13-s − 15-s + 3·17-s + 2·19-s + 6·23-s + 25-s − 5·27-s − 9·29-s + 8·31-s − 3·33-s − 10·37-s + 39-s − 2·43-s + 2·45-s − 3·47-s + 3·51-s + 3·55-s + 2·57-s + 12·59-s − 8·61-s − 65-s − 8·67-s + 6·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 1.43·31-s − 0.522·33-s − 1.64·37-s + 0.160·39-s − 0.304·43-s + 0.298·45-s − 0.437·47-s + 0.420·51-s + 0.404·55-s + 0.264·57-s + 1.56·59-s − 1.02·61-s − 0.124·65-s − 0.977·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.158116040920055066970526248354, −7.50017888966930388217612803697, −6.83102885492469258936209720494, −5.64169956138607318313953641197, −5.25866376062159601464069978070, −4.15029694938806950901022719215, −3.19916988144297977111085735514, −2.78831255508624345418147940865, −1.47337992164748360257213424884, 0,
1.47337992164748360257213424884, 2.78831255508624345418147940865, 3.19916988144297977111085735514, 4.15029694938806950901022719215, 5.25866376062159601464069978070, 5.64169956138607318313953641197, 6.83102885492469258936209720494, 7.50017888966930388217612803697, 8.158116040920055066970526248354