| L(s) = 1 | + 2.52·2-s − 2.69·3-s + 4.39·4-s + 0.644·5-s − 6.81·6-s + 6.06·8-s + 4.26·9-s + 1.63·10-s + 2.63·11-s − 11.8·12-s − 0.526·13-s − 1.73·15-s + 6.54·16-s + 0.858·17-s + 10.7·18-s + 7.96·19-s + 2.83·20-s + 6.65·22-s − 8.34·23-s − 16.3·24-s − 4.58·25-s − 1.33·26-s − 3.42·27-s + 5.08·29-s − 4.39·30-s + 1.52·31-s + 4.43·32-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 1.55·3-s + 2.19·4-s + 0.288·5-s − 2.78·6-s + 2.14·8-s + 1.42·9-s + 0.515·10-s + 0.793·11-s − 3.42·12-s − 0.146·13-s − 0.448·15-s + 1.63·16-s + 0.208·17-s + 2.54·18-s + 1.82·19-s + 0.633·20-s + 1.41·22-s − 1.74·23-s − 3.33·24-s − 0.916·25-s − 0.261·26-s − 0.658·27-s + 0.943·29-s − 0.802·30-s + 0.274·31-s + 0.783·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.626327863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.626327863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 0.644T + 5T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + 0.526T + 13T^{2} \) |
| 17 | \( 1 - 0.858T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 5.08T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 7.77T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.04T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 5.74T + 89T^{2} \) |
| 97 | \( 1 - 6.76T + 97T^{2} \) |
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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
−9.597854066749848554764962973341, −7.928595352630169854532846533636, −7.07126443852981021204922840111, −6.31034062704287536764841927216, −5.81551289469762431980491179912, −5.27520584724690741896440225324, −4.39129755135688815929209408488, −3.73504110750308270683415154064, −2.45503839976922751959798170355, −1.13092851177068270967974268811,
1.13092851177068270967974268811, 2.45503839976922751959798170355, 3.73504110750308270683415154064, 4.39129755135688815929209408488, 5.27520584724690741896440225324, 5.81551289469762431980491179912, 6.31034062704287536764841927216, 7.07126443852981021204922840111, 7.928595352630169854532846533636, 9.597854066749848554764962973341
