| L(s) = 1 | + 1.89·2-s + 1.57·4-s + 2.79·5-s + 1.18·7-s − 0.806·8-s + 5.29·10-s + 6.19·11-s + 6.24·13-s + 2.24·14-s − 4.67·16-s + 7.04·17-s − 8.14·19-s + 4.40·20-s + 11.7·22-s − 23-s + 2.83·25-s + 11.8·26-s + 1.86·28-s + 29-s − 1.41·31-s − 7.21·32-s + 13.3·34-s + 3.32·35-s + 1.97·37-s − 15.3·38-s − 2.25·40-s + 1.66·41-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.786·4-s + 1.25·5-s + 0.448·7-s − 0.285·8-s + 1.67·10-s + 1.86·11-s + 1.73·13-s + 0.599·14-s − 1.16·16-s + 1.70·17-s − 1.86·19-s + 0.984·20-s + 2.49·22-s − 0.208·23-s + 0.566·25-s + 2.31·26-s + 0.353·28-s + 0.185·29-s − 0.253·31-s − 1.27·32-s + 2.28·34-s + 0.561·35-s + 0.324·37-s − 2.49·38-s − 0.356·40-s + 0.259·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.505653395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.505653395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 7.04T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 1.97T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 8.99T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 0.494T + 79T^{2} \) |
| 83 | \( 1 + 7.66T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.20T + 97T^{2} \) |
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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
−8.213888038392562379973554918823, −6.88521198378239540276988110180, −6.27392639218716598553620009112, −5.97485818199823275310701299332, −5.32967036141910832989108997724, −4.27495320322054415273719438856, −3.87131424546146092811570355422, −3.05170796120836659706659369722, −1.85529090165589714640286888051, −1.28760725951446778736902557486,
1.28760725951446778736902557486, 1.85529090165589714640286888051, 3.05170796120836659706659369722, 3.87131424546146092811570355422, 4.27495320322054415273719438856, 5.32967036141910832989108997724, 5.97485818199823275310701299332, 6.27392639218716598553620009112, 6.88521198378239540276988110180, 8.213888038392562379973554918823
