| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 0.267·7-s − 8-s + 9-s + 10-s + 12-s + 4.46·13-s − 0.267·14-s − 15-s + 16-s + 1.73·17-s − 18-s + 7·19-s − 20-s + 0.267·21-s − 6.73·23-s − 24-s + 25-s − 4.46·26-s + 27-s + 0.267·28-s + 5·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.101·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 1.23·13-s − 0.0716·14-s − 0.258·15-s + 0.250·16-s + 0.420·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s + 0.0584·21-s − 1.40·23-s − 0.204·24-s + 0.200·25-s − 0.875·26-s + 0.192·27-s + 0.0506·28-s + 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.746136038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746136038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 0.267T + 7T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 0.732T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 8.73T + 97T^{2} \) |
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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
−8.426432853754242984880277711259, −8.024458608735070932512844260838, −7.29661681183445961991519679853, −6.49211100950202013893424228844, −5.69219356766473759082316869917, −4.65103116710337030500424138165, −3.59749806580967497483605375454, −3.08368739579778775462255261820, −1.82625792256567554795106703959, −0.877587963729738438618425742141,
0.877587963729738438618425742141, 1.82625792256567554795106703959, 3.08368739579778775462255261820, 3.59749806580967497483605375454, 4.65103116710337030500424138165, 5.69219356766473759082316869917, 6.49211100950202013893424228844, 7.29661681183445961991519679853, 8.024458608735070932512844260838, 8.426432853754242984880277711259
