| L(s) = 1 | + 1.50·2-s + 0.906·3-s + 0.264·4-s + 5-s + 1.36·6-s − 0.905·7-s − 2.61·8-s − 2.17·9-s + 1.50·10-s − 5.12·11-s + 0.239·12-s − 2.87·13-s − 1.36·14-s + 0.906·15-s − 4.45·16-s + 7.86·17-s − 3.27·18-s + 1.79·19-s + 0.264·20-s − 0.820·21-s − 7.71·22-s − 1.04·23-s − 2.36·24-s + 25-s − 4.33·26-s − 4.69·27-s − 0.239·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.523·3-s + 0.132·4-s + 0.447·5-s + 0.557·6-s − 0.342·7-s − 0.923·8-s − 0.725·9-s + 0.475·10-s − 1.54·11-s + 0.0692·12-s − 0.798·13-s − 0.363·14-s + 0.234·15-s − 1.11·16-s + 1.90·17-s − 0.772·18-s + 0.412·19-s + 0.0591·20-s − 0.179·21-s − 1.64·22-s − 0.218·23-s − 0.483·24-s + 0.200·25-s − 0.849·26-s − 0.903·27-s − 0.0452·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 3 | \( 1 - 0.906T + 3T^{2} \) |
| 7 | \( 1 + 0.905T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 - 0.565T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 + 9.32T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 1.69T + 79T^{2} \) |
| 83 | \( 1 - 0.788T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.830784718070460305564337918887, −7.83579209239046806286458460107, −7.31508894087483930237829670448, −5.84279552520393553132903947894, −5.56983615475725902956075435089, −4.88567414617468708722203054927, −3.50601769392554596935796274568, −3.08668740331440166323918573043, −2.14867960871441445986176243808, 0,
2.14867960871441445986176243808, 3.08668740331440166323918573043, 3.50601769392554596935796274568, 4.88567414617468708722203054927, 5.56983615475725902956075435089, 5.84279552520393553132903947894, 7.31508894087483930237829670448, 7.83579209239046806286458460107, 8.830784718070460305564337918887
