| L(s) = 1 | − 1.41·2-s + 9.12·3-s − 6.00·4-s + 3.29·5-s − 12.8·6-s − 5.15·7-s + 19.7·8-s + 56.3·9-s − 4.65·10-s − 39.6·11-s − 54.8·12-s − 2.48·13-s + 7.27·14-s + 30.0·15-s + 20.1·16-s − 77.2·17-s − 79.5·18-s − 152.·19-s − 19.8·20-s − 47.0·21-s + 55.9·22-s − 23·23-s + 180.·24-s − 114.·25-s + 3.51·26-s + 267.·27-s + 30.9·28-s + ⋯ |
| L(s) = 1 | − 0.498·2-s + 1.75·3-s − 0.751·4-s + 0.294·5-s − 0.876·6-s − 0.278·7-s + 0.873·8-s + 2.08·9-s − 0.147·10-s − 1.08·11-s − 1.31·12-s − 0.0531·13-s + 0.138·14-s + 0.518·15-s + 0.315·16-s − 1.10·17-s − 1.04·18-s − 1.84·19-s − 0.221·20-s − 0.489·21-s + 0.542·22-s − 0.208·23-s + 1.53·24-s − 0.913·25-s + 0.0264·26-s + 1.90·27-s + 0.209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 1.41T + 8T^{2} \) |
| 3 | \( 1 - 9.12T + 27T^{2} \) |
| 5 | \( 1 - 3.29T + 125T^{2} \) |
| 7 | \( 1 + 5.15T + 343T^{2} \) |
| 11 | \( 1 + 39.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.48T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 14.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 403.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 20.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 12.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 229.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 354.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 375.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 547.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 8.01T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−9.572755091956422336513359377730, −8.719487273944372875216314257479, −8.222998488181841402248395943850, −7.52576413911604364583273199887, −6.28537187293540298162495909006, −4.70295008390710116449450958125, −3.97949382398348058893610921651, −2.70124433286938814381210248617, −1.85441405668588152466446227581, 0,
1.85441405668588152466446227581, 2.70124433286938814381210248617, 3.97949382398348058893610921651, 4.70295008390710116449450958125, 6.28537187293540298162495909006, 7.52576413911604364583273199887, 8.222998488181841402248395943850, 8.719487273944372875216314257479, 9.572755091956422336513359377730
