| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s − 3·13-s − 15-s − 7·17-s + 19-s + 21-s + 5·23-s + 25-s − 5·27-s − 5·29-s − 10·31-s − 35-s + 2·37-s − 3·39-s + 2·41-s − 6·43-s + 2·45-s − 6·49-s − 7·51-s + 9·53-s + 57-s + 7·59-s − 4·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.832·13-s − 0.258·15-s − 1.69·17-s + 0.229·19-s + 0.218·21-s + 1.04·23-s + 1/5·25-s − 0.962·27-s − 0.928·29-s − 1.79·31-s − 0.169·35-s + 0.328·37-s − 0.480·39-s + 0.312·41-s − 0.914·43-s + 0.298·45-s − 6/7·49-s − 0.980·51-s + 1.23·53-s + 0.132·57-s + 0.911·59-s − 0.512·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.986344066518000342261132341873, −8.383457982574818980118436160291, −7.46538735173325270822540586447, −6.90070996304036948382018633686, −5.66067580383108606923524721918, −4.83532426240809128297136495442, −3.87965890870913375486612471780, −2.86645041705704269131297649148, −1.93833527591949064825155603372, 0,
1.93833527591949064825155603372, 2.86645041705704269131297649148, 3.87965890870913375486612471780, 4.83532426240809128297136495442, 5.66067580383108606923524721918, 6.90070996304036948382018633686, 7.46538735173325270822540586447, 8.383457982574818980118436160291, 8.986344066518000342261132341873
