| L(s) = 1 | − 3-s + 1.61·7-s − 2·9-s + 0.763·11-s − 4.85·13-s − 0.763·17-s + 5.85·19-s − 1.61·21-s − 8.23·23-s + 5·27-s − 1.38·29-s + 3·31-s − 0.763·33-s + 4.23·37-s + 4.85·39-s − 5.23·41-s − 1.85·43-s + 1.61·47-s − 4.38·49-s + 0.763·51-s + 5.47·53-s − 5.85·57-s + 4.14·59-s − 4.70·61-s − 3.23·63-s − 9.23·67-s + 8.23·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.611·7-s − 0.666·9-s + 0.230·11-s − 1.34·13-s − 0.185·17-s + 1.34·19-s − 0.353·21-s − 1.71·23-s + 0.962·27-s − 0.256·29-s + 0.538·31-s − 0.132·33-s + 0.696·37-s + 0.777·39-s − 0.817·41-s − 0.282·43-s + 0.236·47-s − 0.625·49-s + 0.106·51-s + 0.751·53-s − 0.775·57-s + 0.539·59-s − 0.602·61-s − 0.407·63-s − 1.12·67-s + 0.991·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144801999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144801999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 - 5.47T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.64824961995607438276222837276, −7.02196972942130398463108662581, −6.15354258616454343894897143745, −5.61970296453562844387834001546, −4.94038330424265115831022506001, −4.40531544784867458299058915958, −3.37325117241713319554365943503, −2.55373460995541527467768255172, −1.70399031143269131512794754504, −0.51381759755279641076785231776,
0.51381759755279641076785231776, 1.70399031143269131512794754504, 2.55373460995541527467768255172, 3.37325117241713319554365943503, 4.40531544784867458299058915958, 4.94038330424265115831022506001, 5.61970296453562844387834001546, 6.15354258616454343894897143745, 7.02196972942130398463108662581, 7.64824961995607438276222837276
