| L(s) = 1 | + 0.445·2-s − 3.24·3-s − 1.80·4-s − 5-s − 1.44·6-s + 3.24·7-s − 1.69·8-s + 7.54·9-s − 0.445·10-s − 0.692·11-s + 5.85·12-s + 1.44·14-s + 3.24·15-s + 2.85·16-s + 3.74·17-s + 3.35·18-s − 1.53·19-s + 1.80·20-s − 10.5·21-s − 0.307·22-s − 1.22·23-s + 5.49·24-s + 25-s − 14.7·27-s − 5.85·28-s − 6.07·29-s + 1.44·30-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 1.87·3-s − 0.900·4-s − 0.447·5-s − 0.589·6-s + 1.22·7-s − 0.598·8-s + 2.51·9-s − 0.140·10-s − 0.208·11-s + 1.68·12-s + 0.386·14-s + 0.838·15-s + 0.712·16-s + 0.907·17-s + 0.791·18-s − 0.351·19-s + 0.402·20-s − 2.30·21-s − 0.0656·22-s − 0.255·23-s + 1.12·24-s + 0.200·25-s − 2.83·27-s − 1.10·28-s − 1.12·29-s + 0.263·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 0.692T + 11T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + 0.457T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 0.801T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 9.87T + 71T^{2} \) |
| 73 | \( 1 - 8.05T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−10.08260074819754363975525712483, −8.980806120475066550950734163951, −7.87742937547111830405240015715, −7.15987609639137326358242594432, −5.78516715931268228571650472370, −5.38732873807321509278610472570, −4.57390845294357769008725299808, −3.82014254941757263839934566899, −1.43288330913151538286107058378, 0,
1.43288330913151538286107058378, 3.82014254941757263839934566899, 4.57390845294357769008725299808, 5.38732873807321509278610472570, 5.78516715931268228571650472370, 7.15987609639137326358242594432, 7.87742937547111830405240015715, 8.980806120475066550950734163951, 10.08260074819754363975525712483
