| L(s) = 1 | + 1.53·2-s + 3-s + 0.369·4-s + 3.17·5-s + 1.53·6-s + 0.630·7-s − 2.51·8-s + 9-s + 4.87·10-s − 11-s + 0.369·12-s − 13-s + 0.971·14-s + 3.17·15-s − 4.60·16-s − 0.539·17-s + 1.53·18-s − 0.630·19-s + 1.17·20-s + 0.630·21-s − 1.53·22-s + 1.55·23-s − 2.51·24-s + 5.04·25-s − 1.53·26-s + 27-s + 0.232·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.577·3-s + 0.184·4-s + 1.41·5-s + 0.628·6-s + 0.238·7-s − 0.887·8-s + 0.333·9-s + 1.54·10-s − 0.301·11-s + 0.106·12-s − 0.277·13-s + 0.259·14-s + 0.818·15-s − 1.15·16-s − 0.130·17-s + 0.362·18-s − 0.144·19-s + 0.261·20-s + 0.137·21-s − 0.328·22-s + 0.323·23-s − 0.512·24-s + 1.00·25-s − 0.301·26-s + 0.192·27-s + 0.0440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.022488227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.022488227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 17 | \( 1 + 0.539T + 17T^{2} \) |
| 19 | \( 1 + 0.630T + 19T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 + 8.29T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 + 2.63T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 4.18T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 + 1.84T + 97T^{2} \) |
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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
−11.27940190419586783090614373617, −10.14013759322070192861896112760, −9.378857010281090988590286581196, −8.625406629353577832905762847294, −7.26526037626892743189356102466, −6.09992496845321813506834934804, −5.37583615980297765280886501478, −4.38326156910320272810477700427, −3.06448576870909594423513298164, −2.00486756170920668362163484465,
2.00486756170920668362163484465, 3.06448576870909594423513298164, 4.38326156910320272810477700427, 5.37583615980297765280886501478, 6.09992496845321813506834934804, 7.26526037626892743189356102466, 8.625406629353577832905762847294, 9.378857010281090988590286581196, 10.14013759322070192861896112760, 11.27940190419586783090614373617
