| L(s) = 1 | − 0.356·2-s + 0.508·3-s − 7.87·4-s − 0.0507·5-s − 0.181·6-s + 30.7·7-s + 5.66·8-s − 26.7·9-s + 0.0180·10-s − 19.0·11-s − 4.00·12-s − 20.4·13-s − 10.9·14-s − 0.0258·15-s + 60.9·16-s − 16.3·17-s + 9.54·18-s + 77.8·19-s + 0.399·20-s + 15.6·21-s + 6.81·22-s + 23·23-s + 2.88·24-s − 124.·25-s + 7.30·26-s − 27.3·27-s − 241.·28-s + ⋯ |
| L(s) = 1 | − 0.126·2-s + 0.0979·3-s − 0.984·4-s − 0.00453·5-s − 0.0123·6-s + 1.65·7-s + 0.250·8-s − 0.990·9-s + 0.000572·10-s − 0.523·11-s − 0.0963·12-s − 0.436·13-s − 0.209·14-s − 0.000444·15-s + 0.952·16-s − 0.233·17-s + 0.124·18-s + 0.939·19-s + 0.00446·20-s + 0.162·21-s + 0.0660·22-s + 0.208·23-s + 0.0245·24-s − 0.999·25-s + 0.0551·26-s − 0.194·27-s − 1.63·28-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 | \( 1 + 0.356T + 8T^{2} \) |
| 3 | \( 1 - 0.508T + 27T^{2} \) |
| 5 | \( 1 + 0.0507T + 125T^{2} \) |
| 7 | \( 1 - 30.7T + 343T^{2} \) |
| 11 | \( 1 + 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.8T + 6.85e3T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 66.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 209.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 25.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 202.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 592.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 853.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 205.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 644.T + 9.12e5T^{2} \) |
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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
−9.634364106672096888389606850898, −8.598812982069868851215486285476, −8.162274768467140536998033519930, −7.38866308414921208569335701566, −5.70227745373342542618366147698, −5.09204269119397483302513263183, −4.26489772632304758185044085943, −2.86902193205714973187160447253, −1.46433774746949862808110176279, 0,
1.46433774746949862808110176279, 2.86902193205714973187160447253, 4.26489772632304758185044085943, 5.09204269119397483302513263183, 5.70227745373342542618366147698, 7.38866308414921208569335701566, 8.162274768467140536998033519930, 8.598812982069868851215486285476, 9.634364106672096888389606850898
