| L(s) = 1 | − 2-s + 1.80·3-s + 4-s + 0.445·5-s − 1.80·6-s − 8-s + 2.24·9-s − 0.445·10-s − 1.24·11-s + 1.80·12-s − 1.24·13-s + 0.801·15-s + 16-s − 0.445·17-s − 2.24·18-s + 0.445·20-s + 1.24·22-s − 1.80·23-s − 1.80·24-s − 0.801·25-s + 1.24·26-s + 2.24·27-s − 0.801·30-s − 32-s − 2.24·33-s + 0.445·34-s + 2.24·36-s + ⋯ |
| L(s) = 1 | − 2-s + 1.80·3-s + 4-s + 0.445·5-s − 1.80·6-s − 8-s + 2.24·9-s − 0.445·10-s − 1.24·11-s + 1.80·12-s − 1.24·13-s + 0.801·15-s + 16-s − 0.445·17-s − 2.24·18-s + 0.445·20-s + 1.24·22-s − 1.80·23-s − 1.80·24-s − 0.801·25-s + 1.24·26-s + 2.24·27-s − 0.801·30-s − 32-s − 2.24·33-s + 0.445·34-s + 2.24·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.002110400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002110400\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 - 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.24T + T^{2} \) |
| 13 | \( 1 + 1.24T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.445T + T^{2} \) |
| 47 | \( 1 - 1.24T + T^{2} \) |
| 53 | \( 1 - 1.80T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.80T + T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.39052250984998208261469000418, −10.01004310592409514211731123294, −9.254848184169428343740878709725, −8.387050183722462468831537672236, −7.73067948880539300272265776007, −7.09852289495463052425338075446, −5.63226423934626143401753205157, −3.99809265331282871294376617322, −2.53382066237305735866409952275, −2.18569641276333263378569885026,
2.18569641276333263378569885026, 2.53382066237305735866409952275, 3.99809265331282871294376617322, 5.63226423934626143401753205157, 7.09852289495463052425338075446, 7.73067948880539300272265776007, 8.387050183722462468831537672236, 9.254848184169428343740878709725, 10.01004310592409514211731123294, 10.39052250984998208261469000418
