| L(s) = 1 | + 5.44·2-s − 0.176·3-s + 21.6·4-s − 0.961·6-s − 7·7-s + 74.1·8-s − 26.9·9-s + 11·11-s − 3.81·12-s − 66.1·13-s − 38.1·14-s + 230.·16-s − 119.·17-s − 146.·18-s − 20.7·19-s + 1.23·21-s + 59.8·22-s − 121.·23-s − 13.0·24-s − 360.·26-s + 9.53·27-s − 151.·28-s + 173.·29-s − 106.·31-s + 662.·32-s − 1.94·33-s − 651.·34-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.0339·3-s + 2.70·4-s − 0.0654·6-s − 0.377·7-s + 3.27·8-s − 0.998·9-s + 0.301·11-s − 0.0918·12-s − 1.41·13-s − 0.727·14-s + 3.60·16-s − 1.70·17-s − 1.92·18-s − 0.250·19-s + 0.0128·21-s + 0.580·22-s − 1.10·23-s − 0.111·24-s − 2.71·26-s + 0.0679·27-s − 1.02·28-s + 1.11·29-s − 0.615·31-s + 3.65·32-s − 0.0102·33-s − 3.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 5.44T + 8T^{2} \) |
| 3 | \( 1 + 0.176T + 27T^{2} \) |
| 13 | \( 1 + 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 87.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 457.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 69.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 504.T + 9.12e5T^{2} \) |
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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
−8.223921630657637685114773044949, −7.23447324312400227305906308106, −6.54689621540550133947467770211, −5.98223666368624266383798425458, −5.03370368135878327246417458532, −4.46746746027728762318680705869, −3.52274979884469779932003476158, −2.61292148615419274204592260731, −2.01053000918588115546112746730, 0,
2.01053000918588115546112746730, 2.61292148615419274204592260731, 3.52274979884469779932003476158, 4.46746746027728762318680705869, 5.03370368135878327246417458532, 5.98223666368624266383798425458, 6.54689621540550133947467770211, 7.23447324312400227305906308106, 8.223921630657637685114773044949
