| L(s) = 1 | + 2-s + 3-s + 4-s − 1.92·5-s + 6-s − 2.26·7-s + 8-s + 9-s − 1.92·10-s − 11-s + 12-s − 0.873·13-s − 2.26·14-s − 1.92·15-s + 16-s − 3.06·17-s + 18-s + 7.77·19-s − 1.92·20-s − 2.26·21-s − 22-s + 4.91·23-s + 24-s − 1.28·25-s − 0.873·26-s + 27-s − 2.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.862·5-s + 0.408·6-s − 0.854·7-s + 0.353·8-s + 0.333·9-s − 0.609·10-s − 0.301·11-s + 0.288·12-s − 0.242·13-s − 0.604·14-s − 0.497·15-s + 0.250·16-s − 0.743·17-s + 0.235·18-s + 1.78·19-s − 0.431·20-s − 0.493·21-s − 0.213·22-s + 1.02·23-s + 0.204·24-s − 0.256·25-s − 0.171·26-s + 0.192·27-s − 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832369534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832369534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 13 | \( 1 + 0.873T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 + 0.988T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 0.410T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 + 2.81T + 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.05T + 71T^{2} \) |
| 73 | \( 1 - 1.69T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.244402490212488864344725090711, −7.62033242572521499654609779747, −7.03371239837449205498167549180, −6.31226187114561700906139246659, −5.30996734456653180288387178904, −4.57704493119620543456935790777, −3.71294738684862334978205667393, −3.11944747111452513126741662238, −2.39029975567659190011560029550, −0.836197649367144851647975569010,
0.836197649367144851647975569010, 2.39029975567659190011560029550, 3.11944747111452513126741662238, 3.71294738684862334978205667393, 4.57704493119620543456935790777, 5.30996734456653180288387178904, 6.31226187114561700906139246659, 7.03371239837449205498167549180, 7.62033242572521499654609779747, 8.244402490212488864344725090711
