| L(s) = 1 | + 3.19·5-s − 4.72·7-s − 1.62·11-s + 6.89·13-s − 7.02·17-s + 0.601·19-s + 3.69·23-s + 5.22·25-s + 7.32·29-s + 5.49·31-s − 15.1·35-s + 37-s − 0.125·41-s − 11.6·43-s + 9.32·47-s + 15.3·49-s + 6.83·53-s − 5.20·55-s + 2.92·59-s + 2·61-s + 22.0·65-s + 10.8·67-s + 9.32·71-s + 0.331·73-s + 7.69·77-s + 0.293·79-s − 9.49·83-s + ⋯ |
| L(s) = 1 | + 1.43·5-s − 1.78·7-s − 0.490·11-s + 1.91·13-s − 1.70·17-s + 0.138·19-s + 0.770·23-s + 1.04·25-s + 1.36·29-s + 0.987·31-s − 2.55·35-s + 0.164·37-s − 0.0196·41-s − 1.78·43-s + 1.36·47-s + 2.19·49-s + 0.938·53-s − 0.701·55-s + 0.381·59-s + 0.256·61-s + 2.73·65-s + 1.32·67-s + 1.10·71-s + 0.0387·73-s + 0.876·77-s + 0.0330·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984491365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984491365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 7.02T + 17T^{2} \) |
| 19 | \( 1 - 0.601T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 41 | \( 1 + 0.125T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 - 0.331T + 73T^{2} \) |
| 79 | \( 1 - 0.293T + 79T^{2} \) |
| 83 | \( 1 + 9.49T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−8.864063209465507753991155535423, −8.471111481552979613286956474981, −6.88457795486991320923641963426, −6.47296314448510709263515300902, −6.03376338872421353472604854794, −5.11620797666622956580874026945, −3.94129555546909072275074497655, −3.00597659177097025759381076414, −2.26661215137658693505206107340, −0.886206285444425301741639805532,
0.886206285444425301741639805532, 2.26661215137658693505206107340, 3.00597659177097025759381076414, 3.94129555546909072275074497655, 5.11620797666622956580874026945, 6.03376338872421353472604854794, 6.47296314448510709263515300902, 6.88457795486991320923641963426, 8.471111481552979613286956474981, 8.864063209465507753991155535423
