| L(s) = 1 | − 4.42·2-s − 3·3-s + 11.5·4-s + 2.84·5-s + 13.2·6-s + 31.6·7-s − 15.8·8-s + 9·9-s − 12.6·10-s − 11·11-s − 34.7·12-s + 5.15·13-s − 140.·14-s − 8.54·15-s − 22.6·16-s + 121.·17-s − 39.8·18-s + 34.8·19-s + 32.9·20-s − 95.0·21-s + 48.6·22-s + 116.·23-s + 47.4·24-s − 116.·25-s − 22.7·26-s − 27·27-s + 366.·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.44·4-s + 0.254·5-s + 0.903·6-s + 1.71·7-s − 0.699·8-s + 0.333·9-s − 0.398·10-s − 0.301·11-s − 0.835·12-s + 0.109·13-s − 2.67·14-s − 0.147·15-s − 0.353·16-s + 1.73·17-s − 0.521·18-s + 0.420·19-s + 0.368·20-s − 0.988·21-s + 0.471·22-s + 1.05·23-s + 0.403·24-s − 0.935·25-s − 0.171·26-s − 0.192·27-s + 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6207980817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6207980817\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.42T + 8T^{2} \) |
| 5 | \( 1 - 2.84T + 125T^{2} \) |
| 7 | \( 1 - 31.6T + 343T^{2} \) |
| 13 | \( 1 - 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−16.81831671504779142087347297125, −15.43766405448659409016357141234, −13.99363146756414607205109330575, −11.93654204009773310562450625040, −10.95980477158803958093093921586, −9.918859945941867237824488883979, −8.380222906896667839992074325438, −7.38687948636624159398935100409, −5.27021583651063597511424464986, −1.40365882153410304370441844951,
1.40365882153410304370441844951, 5.27021583651063597511424464986, 7.38687948636624159398935100409, 8.380222906896667839992074325438, 9.918859945941867237824488883979, 10.95980477158803958093093921586, 11.93654204009773310562450625040, 13.99363146756414607205109330575, 15.43766405448659409016357141234, 16.81831671504779142087347297125
