| L(s) = 1 | − 1.05·2-s + 2.12·3-s − 6.87·4-s − 5·5-s − 2.25·6-s + 5.16·7-s + 15.7·8-s − 22.4·9-s + 5.29·10-s + 11·11-s − 14.6·12-s + 27.0·13-s − 5.47·14-s − 10.6·15-s + 38.3·16-s − 47.6·17-s + 23.8·18-s − 19·19-s + 34.3·20-s + 10.9·21-s − 11.6·22-s + 97.5·23-s + 33.5·24-s + 25·25-s − 28.7·26-s − 105.·27-s − 35.5·28-s + ⋯ |
| L(s) = 1 | − 0.374·2-s + 0.409·3-s − 0.859·4-s − 0.447·5-s − 0.153·6-s + 0.279·7-s + 0.696·8-s − 0.832·9-s + 0.167·10-s + 0.301·11-s − 0.351·12-s + 0.578·13-s − 0.104·14-s − 0.183·15-s + 0.598·16-s − 0.680·17-s + 0.311·18-s − 0.229·19-s + 0.384·20-s + 0.114·21-s − 0.112·22-s + 0.884·23-s + 0.285·24-s + 0.200·25-s − 0.216·26-s − 0.750·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.05T + 8T^{2} \) |
| 3 | \( 1 - 2.12T + 27T^{2} \) |
| 7 | \( 1 - 5.16T + 343T^{2} \) |
| 13 | \( 1 - 27.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 97.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 32.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 37.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 479.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 502.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 922.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 221.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−9.134197889504269165275554074929, −8.274269755424168868824326532337, −7.900290903923939438662360402607, −6.68269158910968664071306271939, −5.61641571704135857230203834119, −4.59244098226981556068110785029, −3.82045700345685897479334455596, −2.71303780669734624920831953000, −1.21704487375011966483924233447, 0,
1.21704487375011966483924233447, 2.71303780669734624920831953000, 3.82045700345685897479334455596, 4.59244098226981556068110785029, 5.61641571704135857230203834119, 6.68269158910968664071306271939, 7.900290903923939438662360402607, 8.274269755424168868824326532337, 9.134197889504269165275554074929
