| L(s) = 1 | + 0.546·3-s − 2.70·9-s − 1.70·11-s − 0.546·13-s + 3.11·17-s + 4.75·19-s − 5.40·23-s − 3.11·27-s − 0.298·29-s − 7.32·31-s − 0.929·33-s − 5.40·37-s − 0.298·39-s + 7.32·41-s + 4·43-s + 8.25·47-s + 1.70·51-s − 1.40·53-s + 2.59·57-s + 7.70·59-s − 9.89·61-s + 8·67-s − 2.95·69-s − 5.40·71-s − 11.3·73-s + 11.1·79-s + 6.40·81-s + ⋯ |
| L(s) = 1 | + 0.315·3-s − 0.900·9-s − 0.513·11-s − 0.151·13-s + 0.755·17-s + 1.09·19-s − 1.12·23-s − 0.599·27-s − 0.0554·29-s − 1.31·31-s − 0.161·33-s − 0.888·37-s − 0.0477·39-s + 1.14·41-s + 0.609·43-s + 1.20·47-s + 0.238·51-s − 0.192·53-s + 0.343·57-s + 1.00·59-s − 1.26·61-s + 0.977·67-s − 0.355·69-s − 0.641·71-s − 1.33·73-s + 1.24·79-s + 0.711·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728161750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728161750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.546T + 3T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 0.546T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 + 0.298T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−7.68119016363860488573489885357, −7.24099413277676328027490242551, −6.13724292710628349392811168541, −5.60539262997090498571477541411, −5.11320351543683278825446588582, −4.02313333403403006430910951232, −3.37838304544697976849788569111, −2.65008767188382146097879888154, −1.85039770767340268923688672711, −0.59640830909633933612180879670,
0.59640830909633933612180879670, 1.85039770767340268923688672711, 2.65008767188382146097879888154, 3.37838304544697976849788569111, 4.02313333403403006430910951232, 5.11320351543683278825446588582, 5.60539262997090498571477541411, 6.13724292710628349392811168541, 7.24099413277676328027490242551, 7.68119016363860488573489885357
