| L(s) = 1 | + 2-s − 3-s + 4-s + 0.504·5-s − 6-s + 2.39·7-s + 8-s + 9-s + 0.504·10-s + 1.10·11-s − 12-s + 2.49·13-s + 2.39·14-s − 0.504·15-s + 16-s − 17-s + 18-s + 8.61·19-s + 0.504·20-s − 2.39·21-s + 1.10·22-s − 0.297·23-s − 24-s − 4.74·25-s + 2.49·26-s − 27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.225·5-s − 0.408·6-s + 0.904·7-s + 0.353·8-s + 0.333·9-s + 0.159·10-s + 0.334·11-s − 0.288·12-s + 0.690·13-s + 0.639·14-s − 0.130·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.97·19-s + 0.112·20-s − 0.522·21-s + 0.236·22-s − 0.0619·23-s − 0.204·24-s − 0.949·25-s + 0.488·26-s − 0.192·27-s + 0.452·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.619366065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.619366065\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.504T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 19 | \( 1 - 8.61T + 19T^{2} \) |
| 23 | \( 1 + 0.297T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 0.580T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 61 | \( 1 - 2.02T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 3.30T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−7.998181575991928413885015684117, −7.16967473061526500694842277048, −6.59063039050797853625686946201, −5.73557621351905095462039395553, −5.26870630174463582963671342366, −4.56413538828941228747334342895, −3.78493820779962529365181867493, −2.91010956908875973244334837370, −1.75400090188036055137575818394, −1.01546051318070919505274825744,
1.01546051318070919505274825744, 1.75400090188036055137575818394, 2.91010956908875973244334837370, 3.78493820779962529365181867493, 4.56413538828941228747334342895, 5.26870630174463582963671342366, 5.73557621351905095462039395553, 6.59063039050797853625686946201, 7.16967473061526500694842277048, 7.998181575991928413885015684117
