| L(s) = 1 | − 2-s + 1.71·3-s + 4-s + 1.76·5-s − 1.71·6-s − 3.74·7-s − 8-s − 0.0653·9-s − 1.76·10-s + 1.71·12-s − 6.06·13-s + 3.74·14-s + 3.02·15-s + 16-s − 0.746·17-s + 0.0653·18-s + 5.68·19-s + 1.76·20-s − 6.42·21-s − 23-s − 1.71·24-s − 1.88·25-s + 6.06·26-s − 5.25·27-s − 3.74·28-s + 5.63·29-s − 3.02·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.989·3-s + 0.5·4-s + 0.789·5-s − 0.699·6-s − 1.41·7-s − 0.353·8-s − 0.0217·9-s − 0.558·10-s + 0.494·12-s − 1.68·13-s + 1.00·14-s + 0.780·15-s + 0.250·16-s − 0.181·17-s + 0.0154·18-s + 1.30·19-s + 0.394·20-s − 1.40·21-s − 0.208·23-s − 0.349·24-s − 0.376·25-s + 1.18·26-s − 1.01·27-s − 0.708·28-s + 1.04·29-s − 0.552·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577504462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577504462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 + 0.746T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 - 0.984T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 + 7.01T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−8.209589239361170798566150515000, −7.48247765887985818177126989337, −6.94787161204776169819194096752, −6.09712423043878201523054620917, −5.50532110169885544314345260152, −4.36561169154037641847744382412, −3.20507746100666021432913150870, −2.72002536104389493515821772791, −2.14216630504595275440977798447, −0.67390441308271180097919293302,
0.67390441308271180097919293302, 2.14216630504595275440977798447, 2.72002536104389493515821772791, 3.20507746100666021432913150870, 4.36561169154037641847744382412, 5.50532110169885544314345260152, 6.09712423043878201523054620917, 6.94787161204776169819194096752, 7.48247765887985818177126989337, 8.209589239361170798566150515000
