| L(s) = 1 | + 2-s + 3-s + 4-s − 1.58·5-s + 6-s + 4.52·7-s + 8-s + 9-s − 1.58·10-s − 2.99·11-s + 12-s + 6.40·13-s + 4.52·14-s − 1.58·15-s + 16-s − 1.56·17-s + 18-s − 19-s − 1.58·20-s + 4.52·21-s − 2.99·22-s + 4.20·23-s + 24-s − 2.48·25-s + 6.40·26-s + 27-s + 4.52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.709·5-s + 0.408·6-s + 1.71·7-s + 0.353·8-s + 0.333·9-s − 0.501·10-s − 0.902·11-s + 0.288·12-s + 1.77·13-s + 1.20·14-s − 0.409·15-s + 0.250·16-s − 0.380·17-s + 0.235·18-s − 0.229·19-s − 0.354·20-s + 0.987·21-s − 0.637·22-s + 0.876·23-s + 0.204·24-s − 0.497·25-s + 1.25·26-s + 0.192·27-s + 0.855·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.645892348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.645892348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 - 4.52T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 59 | \( 1 - 9.31T + 59T^{2} \) |
| 61 | \( 1 + 2.96T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 - 4.78T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 - 7.95T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−7.889470295192960911122161953699, −7.66902439019401594393407568740, −6.69674988982935911145168769043, −5.72538840388958660009732738468, −5.13955926152995238337141067735, −4.20276388569284577722570731215, −3.92921255615444594702128164365, −2.84574543830064515651199402576, −1.99073969422985338808499050277, −1.06644112837339972644064715416,
1.06644112837339972644064715416, 1.99073969422985338808499050277, 2.84574543830064515651199402576, 3.92921255615444594702128164365, 4.20276388569284577722570731215, 5.13955926152995238337141067735, 5.72538840388958660009732738468, 6.69674988982935911145168769043, 7.66902439019401594393407568740, 7.889470295192960911122161953699
