| L(s) = 1 | + 0.864·2-s + 3-s − 1.25·4-s + 2.49·5-s + 0.864·6-s − 2.81·8-s + 9-s + 2.15·10-s − 3.98·11-s − 1.25·12-s − 2.80·13-s + 2.49·15-s + 0.0715·16-s + 6.98·17-s + 0.864·18-s + 5.98·19-s − 3.12·20-s − 3.44·22-s + 23-s − 2.81·24-s + 1.22·25-s − 2.42·26-s + 27-s − 4.06·29-s + 2.15·30-s + 3.44·31-s + 5.68·32-s + ⋯ |
| L(s) = 1 | + 0.611·2-s + 0.577·3-s − 0.626·4-s + 1.11·5-s + 0.353·6-s − 0.994·8-s + 0.333·9-s + 0.682·10-s − 1.20·11-s − 0.361·12-s − 0.778·13-s + 0.644·15-s + 0.0178·16-s + 1.69·17-s + 0.203·18-s + 1.37·19-s − 0.698·20-s − 0.734·22-s + 0.208·23-s − 0.574·24-s + 0.245·25-s − 0.476·26-s + 0.192·27-s − 0.754·29-s + 0.394·30-s + 0.618·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.090995580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.090995580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.864T + 2T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 5.98T + 19T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 - 8.30T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 - 4.46T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 - 3.30T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.729641305338149527041835286594, −7.72392783565573444530232390406, −7.36656567578889421375351410419, −5.95251836399160357071498065104, −5.48146589722902683875664701947, −4.99596774968749296107695229995, −3.88620784221313909112927630990, −2.98365872291062848153340621852, −2.38804536613833213541134765177, −0.956129374099559479662995589806,
0.956129374099559479662995589806, 2.38804536613833213541134765177, 2.98365872291062848153340621852, 3.88620784221313909112927630990, 4.99596774968749296107695229995, 5.48146589722902683875664701947, 5.95251836399160357071498065104, 7.36656567578889421375351410419, 7.72392783565573444530232390406, 8.729641305338149527041835286594
