| L(s) = 1 | + 3·3-s − 5-s − 2·7-s + 6·9-s − 13-s − 3·15-s − 6·21-s + 23-s + 25-s + 9·27-s + 3·29-s + 3·31-s + 2·35-s + 8·37-s − 3·39-s + 3·41-s + 2·43-s − 6·45-s − 11·47-s − 3·49-s + 14·53-s + 8·59-s + 4·61-s − 12·63-s + 65-s + 4·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 0.755·7-s + 2·9-s − 0.277·13-s − 0.774·15-s − 1.30·21-s + 0.208·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s + 0.538·31-s + 0.338·35-s + 1.31·37-s − 0.480·39-s + 0.468·41-s + 0.304·43-s − 0.894·45-s − 1.60·47-s − 3/7·49-s + 1.92·53-s + 1.04·59-s + 0.512·61-s − 1.51·63-s + 0.124·65-s + 0.488·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.412211373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.412211373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.100839769216016087855330039981, −7.28661716063728325841656502687, −6.83016182725971915864486097946, −5.93710451685549313422518193562, −4.78318488737068688635001569068, −4.10832962517779320291300146581, −3.39719644894833663424924345312, −2.80805181614074607009973896877, −2.11851296381874846358515728701, −0.852697835819937161033263849617,
0.852697835819937161033263849617, 2.11851296381874846358515728701, 2.80805181614074607009973896877, 3.39719644894833663424924345312, 4.10832962517779320291300146581, 4.78318488737068688635001569068, 5.93710451685549313422518193562, 6.83016182725971915864486097946, 7.28661716063728325841656502687, 8.100839769216016087855330039981
