| L(s) = 1 | − 3·3-s + 27.0·7-s + 9·9-s − 1.66·11-s − 18.2·13-s − 39.2·17-s + 157.·19-s − 81.0·21-s + 115.·23-s − 27·27-s − 136.·29-s − 27.8·31-s + 5.00·33-s + 358.·37-s + 54.6·39-s − 30.3·41-s + 212.·43-s − 630.·47-s + 386.·49-s + 117.·51-s + 568.·53-s − 471.·57-s + 209.·59-s + 429.·61-s + 243.·63-s − 775.·67-s − 347.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.45·7-s + 0.333·9-s − 0.0457·11-s − 0.388·13-s − 0.559·17-s + 1.89·19-s − 0.841·21-s + 1.05·23-s − 0.192·27-s − 0.871·29-s − 0.161·31-s + 0.0263·33-s + 1.59·37-s + 0.224·39-s − 0.115·41-s + 0.754·43-s − 1.95·47-s + 1.12·49-s + 0.323·51-s + 1.47·53-s − 1.09·57-s + 0.461·59-s + 0.902·61-s + 0.486·63-s − 1.41·67-s − 0.606·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.378214224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378214224\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 27.0T + 343T^{2} \) |
| 11 | \( 1 + 1.66T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 157.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 630.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 568.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 209.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 429.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 775.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 79.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 271.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 206.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 354.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 529.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
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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
−8.600679646051454852763906708306, −7.58769277925102383547031686184, −7.34660245051147705050748385187, −6.19856415855685810344869171828, −5.21850560970478424485349634809, −4.93361727687883798433438658923, −3.92335396605081613347306698235, −2.70855716070484730901650261145, −1.60295087555233565993620574381, −0.75331728777134877329591022896,
0.75331728777134877329591022896, 1.60295087555233565993620574381, 2.70855716070484730901650261145, 3.92335396605081613347306698235, 4.93361727687883798433438658923, 5.21850560970478424485349634809, 6.19856415855685810344869171828, 7.34660245051147705050748385187, 7.58769277925102383547031686184, 8.600679646051454852763906708306
