| L(s) = 1 | − 2-s + 3-s + 4-s − 3.46·5-s − 6-s + 7-s − 8-s + 9-s + 3.46·10-s + 11-s + 12-s + 2·13-s − 14-s − 3.46·15-s + 16-s + 3.46·17-s − 18-s + 5.46·19-s − 3.46·20-s + 21-s − 22-s + 6.92·23-s − 24-s + 6.99·25-s − 2·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.54·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.894·15-s + 0.250·16-s + 0.840·17-s − 0.235·18-s + 1.25·19-s − 0.774·20-s + 0.218·21-s − 0.213·22-s + 1.44·23-s − 0.204·24-s + 1.39·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072179736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072179736\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.17998808305582310956809182042, −10.07863628872376377350248640291, −9.032638248930660592329284262401, −8.317512233996301506091531949146, −7.58015131040833785076560168365, −6.93845055513310391012329955760, −5.28117544514644935282791954773, −3.90942912380034135356675869307, −3.07281231850540375016606100273, −1.12602938014071185532476208364,
1.12602938014071185532476208364, 3.07281231850540375016606100273, 3.90942912380034135356675869307, 5.28117544514644935282791954773, 6.93845055513310391012329955760, 7.58015131040833785076560168365, 8.317512233996301506091531949146, 9.032638248930660592329284262401, 10.07863628872376377350248640291, 11.17998808305582310956809182042
