| L(s) = 1 | − 2.73·2-s − 3·3-s − 0.524·4-s − 21.1·5-s + 8.20·6-s − 25.8·7-s + 23.3·8-s + 9·9-s + 57.7·10-s − 6.96·11-s + 1.57·12-s + 70.7·14-s + 63.3·15-s − 59.5·16-s + 122.·17-s − 24.6·18-s + 43.1·19-s + 11.0·20-s + 77.5·21-s + 19.0·22-s − 75.5·23-s − 69.9·24-s + 321.·25-s − 27·27-s + 13.5·28-s − 163.·29-s − 173.·30-s + ⋯ |
| L(s) = 1 | − 0.966·2-s − 0.577·3-s − 0.0655·4-s − 1.88·5-s + 0.558·6-s − 1.39·7-s + 1.03·8-s + 0.333·9-s + 1.82·10-s − 0.191·11-s + 0.0378·12-s + 1.34·14-s + 1.09·15-s − 0.930·16-s + 1.75·17-s − 0.322·18-s + 0.520·19-s + 0.123·20-s + 0.806·21-s + 0.184·22-s − 0.684·23-s − 0.594·24-s + 2.56·25-s − 0.192·27-s + 0.0915·28-s − 1.04·29-s − 1.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 7 | \( 1 + 25.8T + 343T^{2} \) |
| 11 | \( 1 + 6.96T + 1.33e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.80T + 5.06e4T^{2} \) |
| 41 | \( 1 + 300.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 407.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 536.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 514.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 491.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 762.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 362.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−10.00925390095160190238508076900, −9.268597223037368882041687409276, −8.061867438237991535787360486684, −7.62152027009354848844439497395, −6.70284329623132656232387256372, −5.32008387147971369428495184347, −4.05789802131570440837082861239, −3.29525775143278037016509973139, −0.855618102020396885805666835300, 0,
0.855618102020396885805666835300, 3.29525775143278037016509973139, 4.05789802131570440837082861239, 5.32008387147971369428495184347, 6.70284329623132656232387256372, 7.62152027009354848844439497395, 8.061867438237991535787360486684, 9.268597223037368882041687409276, 10.00925390095160190238508076900
