| L(s) = 1 | + 1.95·2-s − 3·3-s − 4.16·4-s − 14.5·5-s − 5.87·6-s − 14.5·7-s − 23.8·8-s + 9·9-s − 28.5·10-s + 11.2·11-s + 12.4·12-s − 41.1·13-s − 28.3·14-s + 43.6·15-s − 13.3·16-s + 109.·17-s + 17.6·18-s + 47.6·19-s + 60.6·20-s + 43.5·21-s + 21.9·22-s − 23·23-s + 71.4·24-s + 87.0·25-s − 80.4·26-s − 27·27-s + 60.4·28-s + ⋯ |
| L(s) = 1 | + 0.692·2-s − 0.577·3-s − 0.520·4-s − 1.30·5-s − 0.399·6-s − 0.783·7-s − 1.05·8-s + 0.333·9-s − 0.901·10-s + 0.307·11-s + 0.300·12-s − 0.876·13-s − 0.542·14-s + 0.752·15-s − 0.208·16-s + 1.56·17-s + 0.230·18-s + 0.575·19-s + 0.678·20-s + 0.452·21-s + 0.212·22-s − 0.208·23-s + 0.607·24-s + 0.696·25-s − 0.607·26-s − 0.192·27-s + 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 1.95T + 8T^{2} \) |
| 5 | \( 1 + 14.5T + 125T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.6T + 6.85e3T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 289.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 511.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 792.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 4.74T + 3.57e5T^{2} \) |
| 73 | \( 1 - 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 360.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−8.300351263354462139495891394640, −7.48339530990598834231207572924, −6.87003753502062543274645065036, −5.67534205324634728213159521853, −5.28513720902183711818709161897, −4.10683355166834229122456169972, −3.74324493439940680020534552117, −2.80793937773531286001436073513, −0.862455565622712862273855572132, 0,
0.862455565622712862273855572132, 2.80793937773531286001436073513, 3.74324493439940680020534552117, 4.10683355166834229122456169972, 5.28513720902183711818709161897, 5.67534205324634728213159521853, 6.87003753502062543274645065036, 7.48339530990598834231207572924, 8.300351263354462139495891394640
