| L(s) = 1 | + 0.956·5-s − 6.34i·7-s − 14.4i·11-s − 1.76·13-s − 1.34·17-s − 13.0i·19-s + 4.19i·23-s − 24.0·25-s − 38.1·29-s + 0.172i·31-s − 6.07i·35-s − 15.1·37-s + 69.3·41-s + 5.52i·43-s + 66.7i·47-s + ⋯ |
| L(s) = 1 | + 0.191·5-s − 0.906i·7-s − 1.31i·11-s − 0.135·13-s − 0.0790·17-s − 0.685i·19-s + 0.182i·23-s − 0.963·25-s − 1.31·29-s + 0.00557i·31-s − 0.173i·35-s − 0.410·37-s + 1.69·41-s + 0.128i·43-s + 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6827478716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6827478716\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.956T + 25T^{2} \) |
| 7 | \( 1 + 6.34iT - 49T^{2} \) |
| 11 | \( 1 + 14.4iT - 121T^{2} \) |
| 13 | \( 1 + 1.76T + 169T^{2} \) |
| 17 | \( 1 + 1.34T + 289T^{2} \) |
| 19 | \( 1 + 13.0iT - 361T^{2} \) |
| 23 | \( 1 - 4.19iT - 529T^{2} \) |
| 29 | \( 1 + 38.1T + 841T^{2} \) |
| 31 | \( 1 - 0.172iT - 961T^{2} \) |
| 37 | \( 1 + 15.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.52iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 26.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 103.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 68.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 95.6T + 9.40e3T^{2} \) |
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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
−8.843768351531514487521317307894, −7.80355383892809409973513727151, −7.30955482935125879058706652234, −6.21344537125924451355072184932, −5.63501552287139700913627897856, −4.48108242221470790879576807710, −3.68022076482309066831305093072, −2.70627186974806553608593646052, −1.32567585991287015201811595207, −0.17522947364113160425991031003,
1.72527206428198544922296210686, 2.40123787512670366782254058171, 3.68831540002152667137559650804, 4.63680766371289014669198722389, 5.55267553874477322499001110627, 6.19355809725674904485640968318, 7.28144533435889033831658076848, 7.84329629059004576240022482721, 8.916955000291233639171715205600, 9.467716570355245015810541140712
