| L(s) = 1 | − 1.89·2-s + 3-s + 1.58·4-s + 0.892·5-s − 1.89·6-s − 7-s + 0.783·8-s + 9-s − 1.69·10-s − 5.27·11-s + 1.58·12-s + 1.89·14-s + 0.892·15-s − 4.65·16-s + 4.75·17-s − 1.89·18-s + 0.639·19-s + 1.41·20-s − 21-s + 9.99·22-s + 4.64·23-s + 0.783·24-s − 4.20·25-s + 27-s − 1.58·28-s − 3.84·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.577·3-s + 0.793·4-s + 0.399·5-s − 0.773·6-s − 0.377·7-s + 0.276·8-s + 0.333·9-s − 0.534·10-s − 1.59·11-s + 0.457·12-s + 0.506·14-s + 0.230·15-s − 1.16·16-s + 1.15·17-s − 0.446·18-s + 0.146·19-s + 0.316·20-s − 0.218·21-s + 2.13·22-s + 0.969·23-s + 0.159·24-s − 0.840·25-s + 0.192·27-s − 0.299·28-s − 0.714·29-s − 0.308·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9730422020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9730422020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 5 | \( 1 - 0.892T + 5T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 - 0.639T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 - 0.645T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 7.04T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 0.832T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 - 9.57T + 73T^{2} \) |
| 79 | \( 1 + 0.136T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.50T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−8.540102466551755913998982488778, −7.87993210277412583783148036181, −7.53200225251327355645500155727, −6.64971878422027501440298329551, −5.57915532206786499403723215915, −4.90173956059810341306842612917, −3.62871755309855251058453702224, −2.71139688755497541935429975934, −1.89445839593021344095608396491, −0.68538453661684091483890945868,
0.68538453661684091483890945868, 1.89445839593021344095608396491, 2.71139688755497541935429975934, 3.62871755309855251058453702224, 4.90173956059810341306842612917, 5.57915532206786499403723215915, 6.64971878422027501440298329551, 7.53200225251327355645500155727, 7.87993210277412583783148036181, 8.540102466551755913998982488778
