| L(s) = 1 | − 2·2-s − 2.39·3-s + 4·4-s − 3.28·5-s + 4.79·6-s + 33.9·7-s − 8·8-s − 21.2·9-s + 6.56·10-s − 14.5·11-s − 9.58·12-s − 86.5·13-s − 67.8·14-s + 7.86·15-s + 16·16-s − 102.·17-s + 42.5·18-s + 105.·19-s − 13.1·20-s − 81.3·21-s + 29.1·22-s − 135.·23-s + 19.1·24-s − 114.·25-s + 173.·26-s + 115.·27-s + 135.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.461·3-s + 0.5·4-s − 0.293·5-s + 0.326·6-s + 1.83·7-s − 0.353·8-s − 0.787·9-s + 0.207·10-s − 0.399·11-s − 0.230·12-s − 1.84·13-s − 1.29·14-s + 0.135·15-s + 0.250·16-s − 1.46·17-s + 0.556·18-s + 1.27·19-s − 0.146·20-s − 0.845·21-s + 0.282·22-s − 1.22·23-s + 0.163·24-s − 0.913·25-s + 1.30·26-s + 0.824·27-s + 0.916·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6651926607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6651926607\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2.39T + 27T^{2} \) |
| 5 | \( 1 + 3.28T + 125T^{2} \) |
| 7 | \( 1 - 33.9T + 343T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 135.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 613.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 5.15T + 3.57e5T^{2} \) |
| 73 | \( 1 - 428.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 392.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 811.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 11.3T + 9.12e5T^{2} \) |
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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
−8.918348142222592159704761798792, −8.006471012812203214043146598950, −7.70712837725049814865696613707, −6.82939481488752161664465273868, −5.51931550327660311626250096865, −5.11004915821938716912940597743, −4.12593000192918086355929907682, −2.52867322448022524361855152772, −1.89089830385128012099423765203, −0.42487426048351318813295775842,
0.42487426048351318813295775842, 1.89089830385128012099423765203, 2.52867322448022524361855152772, 4.12593000192918086355929907682, 5.11004915821938716912940597743, 5.51931550327660311626250096865, 6.82939481488752161664465273868, 7.70712837725049814865696613707, 8.006471012812203214043146598950, 8.918348142222592159704761798792
