| L(s) = 1 | − 3-s − 3·5-s − 7-s − 2·9-s − 11-s + 13-s + 3·15-s + 4·17-s + 21-s + 5·23-s + 4·25-s + 5·27-s + 4·29-s − 3·31-s + 33-s + 3·35-s − 3·37-s − 39-s + 4·41-s + 10·43-s + 6·45-s + 49-s − 4·51-s − 10·53-s + 3·55-s − 5·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.774·15-s + 0.970·17-s + 0.218·21-s + 1.04·23-s + 4/5·25-s + 0.962·27-s + 0.742·29-s − 0.538·31-s + 0.174·33-s + 0.507·35-s − 0.493·37-s − 0.160·39-s + 0.624·41-s + 1.52·43-s + 0.894·45-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.404·55-s − 0.650·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.959392744074262274245567173484, −7.45924019189179964622227774351, −6.61785780845735807680751548319, −5.82605683836212868696134727362, −5.11504908427535363755010276168, −4.27087566352487773104594524216, −3.38735028172503760223094207084, −2.77367636110590886248302046166, −1.04485962825007948250845014615, 0,
1.04485962825007948250845014615, 2.77367636110590886248302046166, 3.38735028172503760223094207084, 4.27087566352487773104594524216, 5.11504908427535363755010276168, 5.82605683836212868696134727362, 6.61785780845735807680751548319, 7.45924019189179964622227774351, 7.959392744074262274245567173484
